problem stringlengths 1 13.6k | solution stringlengths 0 18.5k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 8
values | synthetic bool 1
class | __index_level_0__ int64 0 742k |
|---|---|---|---|---|---|---|---|---|---|
3.252. $\frac{\cos ^{2}(4 \alpha-3 \pi)-4 \cos ^{2}(2 \alpha-\pi)+3}{\cos ^{2}(4 \alpha+3 \pi)+4 \cos ^{2}(2 \alpha+\pi)-1}$. | Solution.
$$
\begin{aligned}
& \frac{\cos ^{2}(4 \alpha-3 \pi)-4 \cos ^{2}(2 \alpha-\pi)+3}{\cos ^{2}(4 \alpha+3 \pi)+4 \cos ^{2}(2 \alpha+\pi)-1}= \\
& =\frac{(\cos (3 \pi-4 \alpha))^{2}-4(\cos (\pi-2 \alpha))^{2}+3}{(\cos (3 \pi+4 \alpha))^{2}+4(\cos (\pi+2 \alpha))^{2}-1}=\frac{\cos ^{2} 4 \alpha-4 \cos ^{2} 2 \alp... | \operatorname{tg}^{4}2\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,039 |
3.253.
$$
\frac{\cos \left(4 \alpha-\frac{\pi}{2}\right) \sin \left(\frac{5}{2} \pi+2 \alpha\right)}{(1+\cos 2 \alpha)(1+\cos 4 \alpha)}
$$ | Solution.
$$
\begin{aligned}
& \frac{\cos \left(4 \alpha-\frac{\pi}{2}\right) \sin \left(\frac{5}{2} \pi+2 \alpha\right)}{(1+\cos 2 \alpha)(1+\cos 4 \alpha)}=\frac{\cos \left(\frac{\pi}{2}-4 \alpha\right) \sin \left(\frac{5}{2} \pi+2 \alpha\right)}{(1+\cos 2 \alpha)(1+\cos 4 \alpha)}= \\
& =\frac{\sin 4 \alpha \cos 2 ... | \tan\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,040 |
3.254. $4 \cos \left(\alpha-\frac{\pi}{2}\right) \sin ^{3}\left(\frac{\pi}{2}+\alpha\right)-4 \sin \left(\frac{5}{2} \pi-\alpha\right) \cos ^{3}\left(\frac{3}{2} \pi+\alpha\right)$. | Solution.
$$
\begin{aligned}
& 4 \cos \left(\alpha-\frac{\pi}{2}\right) \sin ^{3}\left(\frac{\pi}{2}+\alpha\right)-4 \sin \left(\frac{5}{2} \pi-\alpha\right) \cos ^{3}\left(\frac{3}{2} \pi+\alpha\right)= \\
& =4 \cos \left(\frac{\pi}{2}-\alpha\right)\left(\sin \left(\frac{\pi}{2}+\alpha\right)\right)^{3}-4 \sin \left(... | \sin4\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,041 |
3.255. $\cos ^{4} 2 \alpha-6 \cos ^{2} 2 \alpha \sin ^{2} 2 \alpha+\sin ^{4} 2 \alpha$. | Solution.
$$
\begin{aligned}
& \cos ^{4} 2 \alpha-6 \cos ^{2} 2 \alpha \sin ^{2} 2 \alpha+\sin ^{4} 2 \alpha= \\
& =\cos ^{4} 2 \alpha+\sin ^{4} 2 \alpha-6 \sin ^{2} 2 \alpha \cos ^{2} 2 \alpha= \\
& =\left(\cos ^{2} 2 \alpha+\sin ^{2} 2 \alpha\right)^{2}-2 \sin ^{2} 2 \alpha \cos ^{2} 2 \alpha-6 \sin ^{2} 2 \alpha \c... | \cos8\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,042 |
3.258. $\cos ^{-2} 4 \alpha-\operatorname{tg}^{2}(3 \pi+4 \alpha)-2 \cos ^{2} \alpha-\sqrt{3} \cos \left(\frac{3}{2} \pi-2 \alpha\right)$. | Solution.
$$
\cos ^{-2} 4 \alpha-\operatorname{tg}^{2}(3 \pi+4 \alpha)-2 \cos ^{2} \alpha-\sqrt{3} \cos \left(\frac{3}{2} \pi-2 \alpha\right)=
$$
$$
\begin{aligned}
& =\frac{1}{\cos ^{2} 4 \alpha}-(\operatorname{tg}(3 \pi+4 \alpha))^{2}-2 \cos ^{2} \alpha-\sqrt{3} \cos \left(\frac{3}{2} \pi-2 \alpha\right)= \\
& =\fr... | 2\sin(2\alpha-\frac{\pi}{6}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,044 |
3.259. $\frac{\operatorname{tg}\left(\frac{5}{4} \pi-4 \alpha\right) \sin ^{2}\left(\frac{5}{4} \pi+4 \alpha\right)}{1-2 \cos ^{2} 4 \alpha}$. | Solution.
$$
\begin{aligned}
& \frac{\tan\left(\frac{5}{4} \pi-4 \alpha\right) \sin ^{2}\left(\frac{5}{4} \pi+4 \alpha\right)}{1-2 \cos ^{2} 4 \alpha}=\frac{\tan\left(\frac{5}{4} \pi-4 \alpha\right) \sin ^{2}\left(\frac{5}{4} \pi+4 \alpha\right)}{-\left(2 \cos ^{2} 4 \alpha-1\right)}= \\
& =\left[\tan \frac{x}{2}=\fra... | -\frac{1}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,045 |
3.260. $\frac{4 \sin ^{4}\left(\alpha-\frac{3}{2} \pi\right)}{\sin ^{4}\left(\alpha-\frac{5 \pi}{2}\right)+\cos ^{4}\left(\alpha+\frac{5 \pi}{2}\right)-1}$. | Solution.
$$
\begin{aligned}
& \frac{4 \sin ^{4}\left(\alpha-\frac{3}{2} \pi\right)}{\sin ^{4}\left(\alpha-\frac{5 \pi}{2}\right)+\cos ^{4}\left(\alpha+\frac{5 \pi}{2}\right)-1}= \\
& =\frac{4\left(-\sin \left(\frac{3}{2} \pi-\alpha\right)\right)^{4}}{\left(-\sin \left(\frac{5 \pi}{2}-\alpha\right)\right)^{4}+\left(\c... | -2\operatorname{ctg}^{2}\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,046 |
3.261. $\frac{\sin \left(4 \alpha+\frac{5 \pi}{2}\right)}{1+\cos \left(4 \alpha-\frac{3 \pi}{2}\right)}$.
3.261. $\frac{\sin \left(4 \alpha+\frac{5 \pi}{2}\right)}{1+\cos \left(4 \alpha-\frac{3 \pi}{2}\right)}$.
(Note: The mathematical expression is already in a universal format and does not require translation.) | ## Solution.
$$
\begin{aligned}
& \frac{\sin \left(4 \alpha+\frac{5 \pi}{2}\right)}{1+\cos \left(4 \alpha-\frac{3 \pi}{2}\right)}=\frac{\sin \left(\frac{5 \pi}{2}+4 \alpha\right)}{1+\cos \left(\frac{3 \pi}{2}-4 \alpha\right)}=\frac{\cos 4 \alpha}{1-\sin 4 \alpha}=\frac{\cos 2(2 \alpha)}{1-\sin 2(2 \alpha)}= \\
&= \fra... | \tan(\frac{\pi}{4}+2\alpha) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,047 |
3.262. $\left(\operatorname{tg} 255^{\circ}-\operatorname{tg} 555^{\circ}\right)\left(\operatorname{tg} 795^{\circ}+\operatorname{tg} 195^{\circ}\right)$.
3.262. $\left(\tan 255^{\circ}-\tan 555^{\circ}\right)\left(\tan 795^{\circ}+\tan 195^{\circ}\right)$. | ## Solution.
$$
\begin{aligned}
& \left(\operatorname{tg} 255^{\circ}-\operatorname{tg} 555^{\circ}\right)\left(\operatorname{tg} 795^{\circ}+\operatorname{tg} 195^{\circ}\right)= \\
& =\left(\operatorname{tg}\left(270^{\circ}-15^{\circ}\right)-\operatorname{tg}\left(540^{\circ}+15^{\circ}\right)\right)\left(\operator... | 8\sqrt{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,048 |
3.263. $\frac{\tan 615^{\circ}-\tan 555^{\circ}}{\tan 795^{\circ}+\tan 735^{\circ}}$. | ## Solution.
$$
\begin{aligned}
& \frac{\tan 615^{\circ}-\tan 555^{\circ}}{\tan 795^{\circ}+\tan 735^{\circ}}=\frac{\tan\left(630^{\circ}-15^{\circ}\right)-\tan\left(540^{\circ}+15^{\circ}\right)}{\tan\left(810^{\circ}-15^{\circ}\right)+\tan\left(720^{\circ}+15^{\circ}\right)}=\frac{\cot 15^{\circ}-\tan 15^{\circ}}{\c... | \frac{\sqrt{3}}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,049 |
3.264 .
$$
\frac{\cos \left(2 x+\frac{\pi}{2}\right) \sin \left(\frac{3 \pi}{2}-3 x\right)-\cos (2 x-5 \pi) \cos \left(3 x+\frac{3 \pi}{2}\right)}{\sin \left(\frac{5 \pi}{2}-x\right) \cos 4 x+\sin x \cos \left(\frac{5 \pi}{2}+4 x\right)}
$$ | Solution.
$$
\begin{aligned}
& \frac{\cos \left(2 x+\frac{\pi}{2}\right) \sin \left(\frac{3 \pi}{2}-3 x\right)-\cos (2 x-5 \pi) \cos \left(3 x+\frac{3 \pi}{2}\right)}{\sin \left(\frac{5 \pi}{2}-x\right) \cos 4 x+\sin x \cos \left(\frac{5 \pi}{2}+4 x\right)}= \\
& =\frac{\cos \left(\frac{\pi}{2}+2 x\right) \sin \left(\... | \operatorname{tg}5x | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,050 |
3.265. $\sin (2 x-\pi) \cos (x-3 \pi)+\sin \left(2 x-\frac{9 \pi}{2}\right) \cos \left(x+\frac{\pi}{2}\right)$. | ## Solution.
$$
\begin{aligned}
& \sin (2 x-\pi) \cos (x-3 \pi)+\sin \left(2 x-\frac{9 \pi}{2}\right) \cos \left(x+\frac{\pi}{2}\right)= \\
& =-\sin (\pi-2 x) \cos (3 \pi-x)-\sin \left(4 \pi+\left(\frac{\pi}{2}-2 x\right)\right) \cos \left(\frac{\pi}{2}+x\right)= \\
& =-\sin 2 x(-\cos x)-\cos 2 x(-\sin x)=\sin 2 x \co... | \sin3x | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,051 |
3.266. $\sin (x+2 \pi) \cos \left(2 x-\frac{7 \pi}{2}\right)+\sin \left(\frac{3 \pi}{2}-x\right) \sin \left(2 x-\frac{5 \pi}{2}\right)$. | ## Solution.
$$
\begin{aligned}
& \sin (x+2 \pi) \cos \left(2 x-\frac{7 \pi}{2}\right)+\sin \left(\frac{3 \pi}{2}-x\right) \sin \left(2 x-\frac{5 \pi}{2}\right)= \\
& =\sin (2 \pi+x) \cos \left(\frac{7 \pi}{2}-2 x\right)+\sin \left(\frac{3 \pi}{2}-x\right)\left(-\sin \left(\frac{5 \pi}{2}-2 x\right)\right)= \\
& =\sin... | \cos3x | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,052 |
3.267. $\sqrt{\sin ^{-2}\left(\alpha-\frac{3}{2} \pi\right)+\cos ^{-2}\left(\alpha+\frac{3}{2} \pi\right)}$. | ## Solution.
$$
\begin{aligned}
& \sqrt{\sin ^{-2}\left(\alpha-\frac{3}{2} \pi\right)+\cos ^{-2}\left(\alpha+\frac{3}{2} \pi\right)}= \\
& =\sqrt{\frac{1}{\left(-\sin \left(\frac{3}{2} \pi-\alpha\right)\right)^{2}}+\frac{1}{\left(\cos \left(\frac{3}{2} \pi+\alpha\right)\right)^{2}}}=\sqrt{\frac{1}{\cos ^{2} \alpha}+\f... | 2|\sin^{-1}2\alpha| | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,053 |
3.269. $\frac{3 \cos ^{2}\left(\alpha+270^{\circ}\right)-\sin ^{2}\left(\alpha-270^{\circ}\right)}{3 \sin ^{2}\left(\alpha-90^{\circ}\right)-\cos ^{2}\left(\alpha+90^{\circ}\right)}$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result dire... | Solution.
$$
\begin{aligned}
& \frac{3 \cos ^{2}\left(\alpha+270^{\circ}\right)-\sin ^{2}\left(\alpha-270^{\circ}\right)}{3 \sin ^{2}\left(\alpha-90^{\circ}\right)-\cos ^{2}\left(\alpha+90^{\circ}\right)}= \\
& =\frac{3\left(\cos \left(\alpha+270^{\circ}\right)\right)^{2}-\left(-\sin \left(270^{\circ}-\alpha\right)\ri... | \tan(\alpha+\frac{\pi}{6})\tan(\alpha-\frac{\pi}{6}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,054 |
3.270. $\frac{\sin 2 \alpha+\cos 2 \alpha-\cos 6 \alpha-\sin 6 \alpha}{\sin 4 \alpha+2 \sin ^{2} 2 \alpha-1}$. | Solution.
$$
\begin{aligned}
& \frac{\sin 2 \alpha+\cos 2 \alpha-\cos 6 \alpha-\sin 6 \alpha}{\sin 4 \alpha+2 \sin ^{2} 2 \alpha-1}= \\
& =\frac{(\sin 2 \alpha-\sin 6 \alpha)+(\cos 2 \alpha-\cos 6 \alpha)}{\sin 4 \alpha-\left(1-2 \sin ^{2} 2 \alpha\right)}= \\
& =\left[\sin x-\sin y=2 \cos \frac{x+y}{2} \sin \frac{x-y... | 2\sin2\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,055 |
3.271. $\sqrt{\left(1-\operatorname{tg}^{2} 2 \alpha\right)\left(\operatorname{ctg}^{2} 2 \alpha-1\right)}$.
3.271. $\sqrt{\left(1-\tan^{2} 2 \alpha\right)\left(\cot^{2} 2 \alpha-1\right)}$. | Solution.
$$
\sqrt{\left(1-\operatorname{tg}^{2} 2 \alpha\right)\left(\operatorname{ctg}^{2} 2 \alpha-1\right)}=\sqrt{\left(1-\frac{\sin ^{2} 2 \alpha}{\cos ^{2} 2 \alpha}\right)\left(\frac{\cos ^{2} 2 \alpha}{\sin ^{2} 2 \alpha}-1\right)}=
$$
$$
=\sqrt{\frac{\cos ^{2} 2 \alpha-\sin ^{2} 2 \alpha}{\cos ^{2} 2 \alpha}... | 2|\operatorname{ctg}4\alpha| | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,056 |
3.272. $\frac{\sqrt{\tan \alpha}+\sqrt{\cot \alpha}}{\sqrt{\tan \alpha}-\sqrt{\cot \alpha}}, 0<\alpha<\frac{\pi}{2}$ and $\alpha \neq \frac{\pi}{4}$. | Solution.
$$
\begin{aligned}
& \frac{\sqrt{\tan \alpha}+\sqrt{\cot \alpha}}{\sqrt{\tan \alpha}-\sqrt{\cot \alpha}}=\frac{(\sqrt{\tan \alpha}+\sqrt{\cot \alpha})(\sqrt{\tan \alpha}+\sqrt{\cot \alpha})}{(\sqrt{\tan \alpha}-\sqrt{\cot \alpha})(\sqrt{\tan \alpha}+\sqrt{\cot \alpha})}= \\
& =\frac{(\sqrt{\tan \alpha}+\sqrt... | \cot(\alpha-\frac{\pi}{4}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,057 |
3.273. $\cos ^{6}\left(\alpha-\frac{\pi}{2}\right)+\sin ^{6}\left(\alpha-\frac{3 \pi}{2}\right)-\frac{3}{4}\left(\sin ^{2}\left(\alpha+\frac{\pi}{2}\right)-\cos ^{2}\left(\alpha+\frac{3 \pi}{2}\right)\right)^{2}$. | Solution.
$$
\begin{aligned}
& \cos ^{6}\left(\alpha-\frac{\pi}{2}\right)+\sin ^{6}\left(\alpha-\frac{3 \pi}{2}\right)-\frac{3}{4}\left(\sin ^{2}\left(\alpha+\frac{\pi}{2}\right)-\cos ^{2}\left(\alpha+\frac{3 \pi}{2}\right)\right)^{2}= \\
& =\left(\cos \left(\frac{\pi}{2}-\alpha\right)\right)^{6}+\left(-\sin ^{6}\left... | \frac{1}{4} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,058 |
3.274. $\frac{\sin ^{2} \alpha}{\sin (\alpha-\beta)}+\frac{\sin ^{2} \beta}{\sin (\beta-\alpha)}$. | ## Solution.
$$
\frac{\sin ^{2} \alpha}{\sin (\alpha-\beta)}+\frac{\sin ^{2} \beta}{\sin (\beta-\alpha)}=\frac{\sin ^{2} \alpha}{\sin (\alpha-\beta)}-\frac{\sin ^{2} \beta}{\sin (\alpha-\beta)}=
$$
$$
\begin{aligned}
& =\frac{\frac{1-\cos 2 \alpha}{2}-\frac{1-\cos 2 \beta}{2}}{\sin (\alpha-\beta)}=\frac{\cos 2 \beta-... | \sin(\alpha+\beta) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,059 |
3.275. $\sqrt{\frac{2 \sin \alpha-\sin 2 \alpha}{2 \sin \alpha+\sin 2 \alpha}}$, if: a) $0<\alpha<\pi$; b) $\pi<\alpha<2 \pi$. | ## Solution.
$$
\begin{aligned}
& \sqrt{\frac{2 \sin \alpha - \sin 2 \alpha}{2 \sin \alpha + \sin 2 \alpha}} = \sqrt{\frac{2 \sin \alpha - 2 \sin \alpha \cos \alpha}{2 \sin \alpha + 2 \sin \alpha \cos \alpha}} = \sqrt{\frac{2 \sin \alpha (1 - \cos \alpha)}{2 \sin \alpha (1 + \cos \alpha)}} = \\
& = \sqrt{\frac{1 - \co... | \begin{cases}\tan\frac{\alpha}{2},& | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,060 |
3.276. $\cos ^{2}\left(45^{\circ}+\alpha\right)-\cos ^{2}\left(30^{\circ}-\alpha\right)+\sin 15^{\circ} \sin \left(75^{\circ}-2 \alpha\right)$. | ## Solution.
$$
\cos ^{2}\left(45^{\circ}+\alpha\right)-\cos ^{2}\left(30^{\circ}-\alpha\right)+\sin 15^{\circ} \sin \left(75^{\circ}-2 \alpha\right)=
$$
$$
\begin{aligned}
& =\left[\cos ^{2} \frac{x}{2}=\frac{1+\cos x}{2}, \sin x \sin y=\frac{1}{2}(\cos (x-y)-\cos (x+y))\right]= \\
& =\frac{1+\cos \left(90^{\circ}+2... | -\sin2\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,061 |
3.277. $\sin ^{2}\left(135^{\circ}-2 \alpha\right)-\sin ^{2}\left(210^{\circ}-2 \alpha\right)-\sin 195^{\circ} \cos \left(165^{\circ}-4 \alpha\right)$. | Solution.
$$
\begin{aligned}
& \sin ^{2}\left(135^{\circ}-2 \alpha\right)-\sin ^{2}\left(210^{\circ}-2 \alpha\right)-\sin 195^{\circ} \cos \left(165^{\circ}-4 \alpha\right)= \\
& =\left(\sin \left(180^{\circ}-\left(45^{\circ}+2 \alpha\right)\right)\right)^{2}-\left(\sin \left(180^{\circ}+\left(30^{\circ}-2 \alpha\righ... | \sin4\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,062 |
3.278. $\sqrt{\frac{1+\sin \alpha}{1-\sin \alpha}}-\sqrt{\frac{1-\sin \alpha}{1+\sin \alpha}}$, if: a) $90^{\circ}<\alpha<180^{\circ}$;
b) $270^{\circ}<\alpha<360^{\circ}$. | Solution.
$\sqrt{\frac{1+\sin \alpha}{1-\sin \alpha}}-\sqrt{\frac{1-\sin \alpha}{1+\sin \alpha}}=\frac{(\sqrt{1+\sin \alpha})^{2}-(\sqrt{1-\sin \alpha})^{2}}{\sqrt{(1-\sin \alpha)(1+\sin \alpha)}}=$
$=\frac{1+\sin \alpha-1+\sin \alpha}{\sqrt{1-\sin ^{2} \alpha}}=\frac{2 \sin \alpha}{\sqrt{\cos ^{2} \alpha}}=\frac{2 \... | -2\tan\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,063 |
3.279. $\left(1+\cos \frac{\alpha-3 \pi}{2}\right) \operatorname{ctg} \frac{\pi-\alpha}{4}$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
3.279. $\left(1+\cos \frac{\alpha-3 \pi}{2}\right) \operatorname{ctg} \frac{\pi-... | ## Solution.
$$
\begin{aligned}
& \left(1+\cos \frac{\alpha-3 \pi}{2}\right) \operatorname{ctg} \frac{\pi-\alpha}{4}=\left(1+\cos \left(\frac{\alpha}{2}-\frac{3 \pi}{2}\right)\right) \operatorname{ctg}\left(\frac{\pi}{4}-\frac{\alpha}{4}\right)= \\
& =\left(1+\cos \left(\frac{3 \pi}{2}-\frac{\alpha}{2}\right)\right) \... | \cos\frac{\alpha}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,064 |
3.280. $\frac{\sin ^{2} 4 \alpha+4 \sin ^{4} 2 \alpha-4 \sin ^{2} 2 \alpha \cos ^{2} 2 \alpha}{4-\sin ^{2} 4 \alpha-4 \sin ^{2} 2 \alpha}$. | ## Solution.
$$
\begin{aligned}
& \frac{\sin ^{2} 4 \alpha+4 \sin ^{4} 2 \alpha-4 \sin ^{2} 2 \alpha \cos ^{2} 2 \alpha}{4-\sin ^{2} 4 \alpha-4 \sin ^{2} 2 \alpha}= \\
& =\frac{(\sin 2(2 \alpha))^{2}+4 \sin ^{4} 2 \alpha-4 \sin ^{2} 2 \alpha \cos ^{2} 2 \alpha}{\left(4-4 \sin ^{2} 4 \alpha\right)-(\sin 2(2 \alpha))^{2... | \tan^{4}2\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,065 |
3.281. $\sin \left(\frac{5}{2} \pi+4 \alpha\right)-\sin ^{6}\left(\frac{5}{2} \pi+2 \alpha\right)+\cos ^{6}\left(\frac{7}{2} \pi-2 \alpha\right)$. | ## Solution.
$$
\begin{aligned}
& \sin \left(\frac{5}{2} \pi+4 \alpha\right)-\sin ^{6}\left(\frac{5}{2} \pi+2 \alpha\right)+\cos ^{6}\left(\frac{7}{2} \pi-2 \alpha\right)= \\
& =\sin \left(\frac{5}{2} \pi+4 \alpha\right)-\left(\sin \left(\frac{5}{2} \pi+2 \alpha\right)\right)^{6}+\left(\cos \left(\frac{7}{2} \pi-2 \al... | \frac{1}{8}\sin8\alpha\sin4\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,066 |
3.282. $\frac{\sin 8 \alpha+\sin 9 \alpha+\sin 10 \alpha+\sin 11 \alpha}{\cos 8 \alpha+\cos 9 \alpha+\cos 10 \alpha+\cos 11 \alpha} \times$
$\times \frac{\cos 8 \alpha-\cos 9 \alpha-\cos 10 \alpha+\cos 11 \alpha}{\sin 8 \alpha-\sin 9 \alpha-\sin 10 \alpha+\sin 11 \alpha}$. | Solution.
$$
\begin{aligned}
& \frac{\sin 8 \alpha + \sin 9 \alpha + \sin 10 \alpha + \sin 11 \alpha}{\cos 8 \alpha + \cos 9 \alpha + \cos 10 \alpha + \cos 11 \alpha} \cdot \frac{\cos 8 \alpha - \cos 9 \alpha - \cos 10 \alpha + \cos 11 \alpha}{\sin 8 \alpha - \sin 9 \alpha - \sin 10 \alpha + \sin 11 \alpha} = \\
& = \... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,067 |
3.283. $\cos \left(270^{\circ}-2 \alpha\right) \operatorname{ctg}\left(30^{\circ}-2 \alpha\right) \operatorname{tg}\left(240^{\circ}-2 \alpha\right)(2 \cos 4 \alpha-1)$. | ## Solution.
$$
\begin{aligned}
& \cos \left(270^{\circ}-2 \alpha\right) \operatorname{ctg}\left(30^{\circ}-2 \alpha\right) \operatorname{tg}\left(240^{\circ}-2 \alpha\right)(2 \cos 4 \alpha-1)= \\
& =\cos \left(270^{\circ}-2 \alpha\right) \operatorname{ctg}\left(30^{\circ}-2 \alpha\right) \operatorname{tg}\left(270^{... | -\sin6\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,068 |
3.284. $\operatorname{tg}\left(2 \operatorname{arctg}\left(\frac{1-\cos x}{\sin x}\right)\right) \sqrt{\frac{1+\cos 2 x}{1-\cos 2 x}}$.
3.284. $\tan\left(2 \arctan\left(\frac{1-\cos x}{\sin x}\right)\right) \sqrt{\frac{1+\cos 2 x}{1-\cos 2 x}}$. | ## Solution.
$$
\begin{aligned}
& \operatorname{tg}\left(2 \operatorname{arctg}\left(\frac{1-\cos x}{\sin x}\right)\right) \sqrt{\frac{1+\cos 2 x}{1-\cos 2 x}}= \\
& =\frac{2 \operatorname{tg}\left(\operatorname{arctg} \frac{1-\cos x}{\sin x}\right)}{1-\operatorname{tg}^{2}\left(\operatorname{arctg} \frac{1-\cos x}{\s... | 1,if\operatorname{ctg}x>0;-1,if\operatorname{ctg}x<0 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,069 |
3.285. $\sin 6 \alpha-2 \sqrt{3} \cos ^{2} 3 \alpha+\sqrt{3}$. | Solution.
$$
\begin{aligned}
& \sin 6 \alpha-2 \sqrt{3} \cos ^{2} 3 \alpha+\sqrt{3}=\left[\cos ^{2} \frac{x}{2}=\frac{1+\cos x}{2}\right]= \\
& =\sin 6 \alpha-\sqrt{3} \cos 6 \alpha=2\left(\frac{1}{2} \sin 6 \alpha-\frac{\sqrt{3}}{2} \cos 6 \alpha\right)= \\
& =2\left(\cos 60^{\circ} \sin 6 \alpha-\sin 60^{\circ} \cos... | 2\sin(6\alpha-60) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,070 |
3.286. $\frac{1}{\sqrt{3}} \sin 4 \alpha + 1 - 2 \cos ^{2} 2 \alpha$. | Solution.
$$
\frac{1}{\sqrt{3}} \sin 4 \alpha + 1 - 2 \cos^2 2 \alpha = \frac{1}{\sqrt{3}} \sin 4 \alpha - (2 \cos^2 2 \alpha - 1) =
$$
$= \frac{1}{\sqrt{3}} \sin 4 \alpha - \cos 4 \alpha = \frac{2}{\sqrt{3}} \left( \frac{1}{2} \sin 4 \alpha - \frac{\sqrt{3}}{2} \cos 4 \alpha \right) =$
$= \frac{2}{\sqrt{3}} \left( ... | \frac{2}{\sqrt{3}}\sin(4\alpha-60) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,071 |
3.287. $3-4 \cos 4 \alpha+\cos 8 \alpha-8 \cos ^{4} 2 \alpha$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
3.287. $3-4 \cos 4 \alpha+\cos 8 \alpha-8 \cos ^{4} 2 \alpha$. | ## Solution.
$$
\begin{aligned}
& 3-4 \cos 4 \alpha+\cos 8 \alpha-8 \cos ^{4} 2 \alpha= \\
& =\left[\cos 2 x=2 \cos ^{2} x-1 ; \cos ^{2} \frac{x}{2}=\frac{1+\cos x}{2}\right]= \\
& =3-4 \cos 4 \alpha+2 \cos ^{2} 4 \alpha-1-8\left(\frac{1+\cos 4 \alpha}{2}\right)^{2}= \\
& =2-4 \cos 4 \alpha+2 \cos ^{2} 4 \alpha-2\left... | -8\cos4\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,072 |
3.288. $\operatorname{tg}^{3} x-\operatorname{tg}^{2} x-3 \operatorname{tg} x+3$.
3.288. $\tan^{3} x-\tan^{2} x-3 \tan x+3$. | Solution.
$$
\begin{aligned}
& \operatorname{tg}^{3} x-\operatorname{tg}^{2} x-3 \operatorname{tg} x+3=\operatorname{tg}^{2} x(\operatorname{tg} x-1)-3(\operatorname{tg} x-1)=(\operatorname{tg} x-1)\left(\operatorname{tg}^{2} x-3\right)= \\
& =\left(\frac{\sin x}{\cos x}-1\right)\left(\frac{\sin ^{2} x}{\cos ^{2} x}-3... | \frac{4\sqrt{2}\sin(x-45)\sin(x-60)\sin(x+60)}{\cos^{3}x} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,073 |
3.289. $\operatorname{tg}^{4} x-4 \operatorname{tg}^{2} x+3$.
3.289. $\tan^{4} x-4 \tan^{2} x+3$. | ## Solution.
$$
\begin{aligned}
& \operatorname{tg}^{4} x-4 \operatorname{tg}^{2} x+3=\operatorname{tg}^{4} x-\operatorname{tg}^{2} x-3 \operatorname{tg}^{2} x+3= \\
& =\operatorname{tg}^{2} x\left(\operatorname{tg}^{2} x-1\right)-3\left(\operatorname{tg}^{2} x-1\right)=\left(\operatorname{tg}^{2} x-1\right)\left(\ope... | \frac{8\sin(x-45)\sin(x+45)\sin(x-60)\sin(x+60)}{\cos^{4}x} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,074 |
3.290. $6 \sin ^{2} 2 \alpha-1-\cos 4 \alpha$.
3.290. $6 \sin ^{2} 2 \alpha-1-\cos 4 \alpha$. | Solution.
$$
\begin{aligned}
& 6 \sin ^{2} 2 \alpha-1-\cos 4 \alpha=6 \sin ^{2} 2 \alpha-1-\left(2 \cos ^{2} 2 \alpha-1\right)= \\
& =6 \sin ^{2} 2 \alpha-2 \cos ^{2} 2 \alpha=2\left(3 \sin ^{2} 2 \alpha-\cos ^{2} 2 \alpha\right)= \\
& =2(\sqrt{3} \sin 2 \alpha-\cos 2 \alpha)(\sqrt{3} \sin 2 \alpha+\cos 2 \alpha)= \\
... | -8\cos(2\alpha+60)\cos(2\alpha-60) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,075 |
3.291. $\sqrt{1+\sin \frac{\alpha}{2}}-\sqrt{1-\sin \frac{\alpha}{2}}$, if $0^{\circ}<\alpha \leq 180^{\circ}$. | ## Solution.
$$
\begin{aligned}
& \sqrt{1+\sin \frac{\alpha}{2}}-\sqrt{1-\sin \frac{\alpha}{2}}=\sqrt{1+\sin 2\left(\frac{\alpha}{4}\right)}-\sqrt{1-\sin 2\left(\frac{\alpha}{4}\right)}= \\
& =\sqrt{1+2 \sin \frac{\alpha}{4} \cos \frac{\alpha}{4}}-\sqrt{1-2 \sin \frac{\alpha}{4} \cos \frac{\alpha}{4}}=
\end{aligned}
$... | 2\sin\frac{\alpha}{4} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,076 |
3.292. $2 \cos ^{2} 2 \alpha+3 \cos 4 \alpha-3$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
3.292. $2 \cos ^{2} 2 \alpha+3 \cos 4 \alpha-3$. | Solution.
$$
\begin{aligned}
& 2 \cos ^{2} 2 \alpha+3 \cos 4 \alpha-3=2 \cos ^{2} 2 \alpha+3\left(1-2 \sin ^{2} 2 \alpha\right)-3= \\
& =2 \cos ^{2} 2 \alpha-6 \sin ^{2} 2 \alpha=2\left(\cos ^{2} 2 \alpha-3 \sin ^{2} 2 \alpha\right)= \\
& =2(\cos 2 \alpha-\sqrt{3} \sin 2 \alpha)(\cos 2 \alpha+\sqrt{3} \sin 2 \alpha)= ... | 8\cos(2\alpha+60)\cos(2\alpha-60) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,077 |
3.293. $\cos ^{2} \alpha+\cos ^{2} \beta-2 \cos \alpha \cos \beta \cos (\alpha-\beta)$. | ## Solution.
$$
\begin{aligned}
& \cos ^{2} \alpha+\cos ^{2} \beta-2 \cos \alpha \cos \beta \cos (\alpha-\beta)= \\
& =\frac{1+\cos 2 \alpha}{2}+\frac{1+\cos 2 \beta}{2}-2 \cos \alpha \cos \beta \cos (\alpha-\beta)=
\end{aligned}
$$
$$
\begin{aligned}
& =\frac{1}{2}(2+(\cos 2 \alpha+\cos 2 \beta))-2 \cos \alpha \cos ... | \sin^{2}(\alpha-\beta) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,078 |
3.294. $\frac{\sin (2 \alpha-\beta)}{\cos 4 \alpha}+\frac{\sin \beta}{\cos 2 \alpha}$. | ## Solution.
$$
\begin{aligned}
& \frac{\sin (2 \alpha-\beta)}{\cos 4 \alpha}+\frac{\sin \beta}{\cos 2 \alpha}=\frac{\sin (2 \alpha-\beta) \cos 2 \alpha+\sin \beta \cos 4 \alpha}{\cos 4 \alpha \cos 2 \alpha}= \\
& =\left[\sin x \cos y=\frac{1}{2}(\sin (x-y)+\sin (x+y))\right]= \\
& =\frac{\frac{1}{2}(\sin (-\beta)+\si... | \frac{\cos(2\alpha+\beta)}{\cos4\alpha}\cdot\tan2\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,079 |
3.295. $\frac{\sin ^{2}(\alpha+\beta)-\sin ^{2} \alpha-\sin ^{2} \beta}{\sin ^{2}(\alpha+\beta)-\cos ^{2} \alpha-\cos ^{2} \beta}$. | Solution.
$$
\begin{aligned}
& \frac{\sin ^{2}(\alpha+\beta)-\sin ^{2} \alpha-\sin ^{2} \beta}{\sin ^{2}(\alpha+\beta)-\cos ^{2} \alpha-\cos ^{2} \beta}= \\
& =\left[\sin ^{2} \frac{x}{2}=\frac{1-\cos x}{2} ; \cos ^{2} \frac{x}{2}=\frac{1+\cos x}{2}\right]= \\
& =\frac{1-\cos ^{2}(\alpha+\beta)-\frac{1-\cos 2 \alpha}{... | -\tan\alpha\cdot\tan\beta | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,080 |
3.296. $\sin ^{2}(\alpha-2 \beta)-\cos ^{2} \alpha-\cos ^{2} 2 \beta$.
3.296. $\sin ^{2}(\alpha-2 \beta)-\cos ^{2} \alpha-\cos ^{2} 2 \beta$. | Solution.
$$
\begin{aligned}
& \sin ^{2}(\alpha-2 \beta)-\cos ^{2} \alpha-\cos ^{2} 2 \beta=\left[\cos ^{2} \frac{x}{2}=\frac{1+\cos x}{2}\right]= \\
& =\sin ^{2}(\alpha-2 \beta)=\frac{1+\cos 2 \alpha}{2}-\frac{1+\cos 4 \beta}{2}= \\
& =\sin ^{2}(\alpha-2 \beta)-\frac{1}{2}(2+(\cos 2 \alpha+\cos 4 \beta))= \\
& =\left... | -2\cos\alpha\cos2\beta\cos(\alpha-2\beta) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,081 |
3.297. $\sin ^{2}(2 \alpha-\beta)-\sin ^{2} 2 \alpha-\sin ^{2} \beta$. | ## Solution.
$$
\sin ^{2}(2 \alpha-\beta)-\sin ^{2} 2 \alpha-\sin ^{2} \beta=\left[\sin ^{2} \frac{x}{2}=\frac{1-\cos x}{2}\right]=
$$
$$
\begin{aligned}
& =\sin ^{2}(2 \alpha-\beta)-\frac{1-\cos 4 \alpha}{2}-\frac{1-\cos 2 \beta}{2}= \\
& =\sin ^{2}(2 \alpha-\beta)-\frac{1}{2}(2-(\cos 4 \alpha+\cos 2 \beta))= \\
& =... | -2\sin2\alpha\sin\beta\cos(2\alpha-\beta) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,082 |
3.299. $2-\frac{\sin 8 \alpha}{\sin ^{4} 2 \alpha-\cos ^{4} 2 \alpha}$. | ## Solution.
$$
\begin{aligned}
& 2-\frac{\sin 8 \alpha}{\sin ^{4} 2 \alpha-\cos ^{4} 2 \alpha}=2+\frac{\sin 2(4 \alpha)}{\cos ^{4} 2 \alpha-\sin ^{4} 2 \alpha}= \\
& =2+\frac{2 \sin 4 \alpha \cos 4 \alpha}{\left(\cos ^{2} 2 \alpha-\sin ^{2} 2 \alpha\right)\left(\cos ^{2} 2 \alpha+\sin ^{2} 2 \alpha\right)}=2+\frac{2 ... | 4\cos^{2}(\frac{\pi}{4}-2\alpha) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,084 |
3.300. $2-\operatorname{tg} 4 \alpha-\operatorname{ctg} 4 \alpha$.
3.300. $2-\tan 4 \alpha-\cot 4 \alpha$. | ## Solution.
$$
\begin{aligned}
& 2-\operatorname{tg} 4 \alpha-\operatorname{ctg} 4 \alpha=2-\left(\frac{\sin 4 \alpha}{\cos 4 \alpha}+\frac{\cos 4 \alpha}{\sin 4 \alpha}\right)=2-\frac{\sin ^{2} 4 \alpha+\cos ^{2} 4 \alpha}{\sin 4 \alpha \cos 4 \alpha}= \\
& =2-\frac{1}{\sin 4 \alpha \cos 4 \alpha}=2-\frac{2}{2 \sin ... | \frac{-4\sin^{2}(\frac{\pi}{4}-4\alpha)}{\sin8\alpha} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,085 |
3.301. $\frac{2 \cos ^{2} 2 \alpha-1}{2 \operatorname{tg}\left(\frac{\pi}{4}-2 \alpha\right) \sin ^{2}\left(\frac{3}{4} \pi-2 \alpha\right)}-\operatorname{tg} 2 \alpha+\cos 2 \alpha-\sin 2 \alpha$. | ## Solution.
$$
\begin{aligned}
& \frac{2 \cos ^{2} 2 \alpha-1}{2 \operatorname{tg}\left(\frac{\pi}{4}-2 \alpha\right) \sin ^{2}\left(\frac{3}{4} \pi-2 \alpha\right)}-\operatorname{tg} 2 \alpha+\cos 2 \alpha-\sin 2 \alpha= \\
& =\left[2 \cos ^{2} 2 x-1=\cos 4 x ; \operatorname{tg} \frac{x}{2}=\frac{\sin x}{1+\cos x}\r... | 2\operatorname{ctg}4\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,086 |
3.303. $1-\sin ^{2} \alpha-\sin ^{2} \beta+2 \sin \alpha \sin \beta \cos (\alpha-\beta)$. | ## Solution.
$$
\begin{aligned}
& 1-\sin ^{2} \alpha-\sin ^{2} \beta+2 \sin \alpha \sin \beta \cos (\alpha-\beta)= \\
& =1-\frac{1-\cos 2 \alpha}{2}-\frac{1-\cos 2 \beta}{2}+2 \sin \alpha \sin \beta \cos (\alpha-\beta)= \\
& =\frac{1}{2}(\cos 2 \alpha+\cos 2 \beta)+2 \sin \alpha \sin \beta \cos (\alpha-\beta)= \\
& =\... | \cos^{2}(\alpha-\beta) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,087 |
3.304. $1+\cos \left(2 \alpha-\frac{3}{2} \pi\right)+\sin \left(2 \alpha+\frac{3}{2} \pi\right)-\operatorname{ctg}\left(\frac{\pi}{2}+2 \alpha\right)$. | ## Solution.
$$
1+\cos \left(2 \alpha-\frac{3}{2} \pi\right)+\sin \left(2 \alpha+\frac{3}{2} \pi\right)-\operatorname{ctg}\left(\frac{\pi}{2}+2 \alpha\right)=
$$
$$
\begin{aligned}
& =1+\cos \left(\frac{3}{2} \pi-2 \alpha\right)+\sin \left(\frac{3}{2} \pi+2 \alpha\right)-\operatorname{ctg}\left(\frac{\pi}{2}+2 \alpha... | \frac{2\sqrt{2}\sin^{2}\alpha\cos(\frac{\pi}{4}-2\alpha)}{\cos2\alpha} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,088 |
3.306. $\frac{\sqrt{1+\sin \alpha}+\sqrt{1-\sin \alpha}}{\sqrt{1+\sin \alpha}-\sqrt{1-\sin \alpha}}$, if a) $0^{\circ}<\alpha<90^{\circ}$; b) $90^{\circ}<\alpha<180^{\circ}$. | ## Solution.
From the condition we have
$$
\begin{aligned}
& \frac{(\sqrt{1+\sin \alpha}+\sqrt{1-\sin \alpha})(\sqrt{1+\sin \alpha}+\sqrt{1-\sin \alpha})}{(\sqrt{1+\sin \alpha}-\sqrt{1-\sin \alpha})(\sqrt{1+\sin \alpha}+\sqrt{1-\sin \alpha})}= \\
& =\frac{(\sqrt{1+\sin \alpha}+\sqrt{1-\sin \alpha})^{2}}{(\sqrt{1+\sin... | )\operatorname{ctg}\frac{\alpha}{2};b)\operatorname{tg}\frac{\alpha}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,089 |
3.307. $2 \sin ^{2} 2 \alpha+\sqrt{3} \sin 4 \alpha-\frac{4 \tan 2 \alpha\left(1-\tan^{2} 2 \alpha\right)}{\sin 8 \alpha\left(1+\tan^{2} 2 \alpha\right)^{2}}$. | Solution.
$$
\begin{aligned}
& 2 \sin ^{2} 2 \alpha+\sqrt{3} \sin 4 \alpha-\frac{4 \operatorname{tg} 2 \alpha\left(1-\operatorname{tg}^{2} 2 \alpha\right)}{\sin 8 \alpha\left(1+\operatorname{tg}^{2} 2 \alpha\right)^{2}}= \\
& =2 \sin ^{2} 2 \alpha+\sqrt{3} \sin 4 \alpha-\frac{2 \operatorname{tg} 2 \alpha}{1+\operatorn... | 2\sin(4\alpha-\frac{\pi}{6}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,090 |
3.308. $\cos ^{2}(\alpha-2 \beta)-\cos ^{2}\left(\alpha-\frac{\pi}{2}\right)-\cos ^{2}(2 \beta-\pi)$. | Solution.
$$
\begin{aligned}
& \cos ^{2}(\alpha-2 \beta)-\cos ^{2}\left(\alpha-\frac{\pi}{2}\right)-\cos ^{2}(2 \beta-\pi)= \\
& =\cos ^{2}(\alpha-2 \beta)-\cos ^{2}\left(\frac{\pi}{2}-\alpha\right)-\cos ^{2}(\pi-2 \beta)=\left[\cos ^{2} \frac{x}{2}=\frac{1+\cos x}{2}\right]= \\
& =\frac{1+\cos (2 \alpha-4 \beta)}{2}-... | 2\sin\alpha\sin(2\beta-\alpha)\cos2\beta | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,091 |
3.309. $1-\cos (\pi-8 \alpha)-\cos (\pi+4 \alpha)$. | ## Solution.
$$
\begin{aligned}
& 1-\cos (\pi-8 \alpha)-\cos (\pi+4 \alpha)=1+\cos 8 \alpha+\cos 4 \alpha= \\
& =1+\cos 2(4 \alpha)+\cos 4 \alpha=1+2 \cos ^{2} 4 \alpha-1+\cos 4 \alpha=2 \cos ^{2} 4 \alpha+\cos 4 \alpha= \\
& =2 \cos 4 \alpha\left(\cos 4 \alpha+\frac{1}{2}\right)=2 \cos 4 \alpha\left(\cos 4 \alpha+\co... | 4\cos4\alpha\cos(2\alpha+\frac{\pi}{6})\cos(2\alpha-\frac{\pi}{6}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,092 |
3.310. $\cos 2 \alpha-\sin 4 \alpha-\cos 6 \alpha$. | ## Solution.
$\cos 2 \alpha-\sin 4 \alpha-\cos 6 \alpha=(\cos 2 \alpha-\cos 6 \alpha)-\sin 4 \alpha=$ $=\left[\cos x-\cos y=-2 \sin \frac{x+y}{2} \sin \frac{x-y}{2}\right]=-2 \sin 4 \alpha \sin (-2 \alpha)-\sin 4 \alpha=$ $=2 \sin 4 \alpha \sin 2 \alpha-\sin 4 \alpha=2 \sin 4 \alpha\left(\sin 2 \alpha-\frac{1}{2}\righ... | 4\sin4\alpha\sin(\alpha-15)\cos(\alpha+15) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,093 |
3.311. $\sin ^{3}\left(\frac{\pi}{2}+\alpha\right)+\cos ^{3}\left(\frac{\pi}{2}+\alpha\right)-\cos \left(\alpha-\frac{3}{2} \pi\right)+\sin \left(\frac{3}{2} \pi+\alpha\right)$. | Solution.
$$
\begin{aligned}
& \sin ^{3}\left(\frac{\pi}{2}+\alpha\right)+\cos ^{3}\left(\frac{\pi}{2}+\alpha\right)-\cos \left(\alpha-\frac{3}{2} \pi\right)+\sin \left(\frac{3}{2} \pi+\alpha\right)= \\
& =\left(\sin \left(\frac{\pi}{2}+\alpha\right)\right)^{3}+\left(\cos \left(\frac{\pi}{2}+\alpha\right)\right)^{3}-\... | \frac{\sqrt{2}}{2}\sin2\alpha\cos(\frac{\pi}{4}+\alpha) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,094 |
3.312. $2 \cos ^{2} 2 \alpha+\sqrt{3} \sin 4 \alpha-1$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
3.312. $2 \cos ^{2} 2 \alpha+\sqrt{3} \sin 4 \alpha-1$. | Solution.
$$
\begin{aligned}
& 2 \cos ^{2} 2 \alpha+\sqrt{3} \sin 4 \alpha-1=\left[\cos ^{2} \frac{x}{2}=\frac{1+\cos x}{2}\right]= \\
& =1+\cos 4 \alpha+\sqrt{3} \sin 4 \alpha-1= \\
& =\cos 4 \alpha+\sqrt{3} \sin 4 \alpha=2\left(\frac{1}{2} \cos 4 \alpha+\frac{\sqrt{3}}{2} \sin 4 \alpha\right)= \\
& =2\left(\cos \fra... | 2\cos(\frac{\pi}{3}-4\alpha) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,095 |
3.314. $\frac{\cos 2 \alpha-\sin 4 \alpha-\cos 6 \alpha}{\cos 2 \alpha+\sin 4 \alpha-\cos 6 \alpha}$. | ## Solution.
$$
\begin{aligned}
& \frac{\cos 2 \alpha - \sin 4 \alpha - \cos 6 \alpha}{\cos 2 \alpha + \sin 4 \alpha - \cos 6 \alpha} = \frac{(\cos 2 \alpha - \cos 6 \alpha) - \sin 4 \alpha}{(\cos 2 \alpha - \cos 6 \alpha) + \sin 4 \alpha} = \\
& = \left[\cos x - \cos y = -2 \sin \frac{x+y}{2} \sin \frac{x-y}{2}\right... | \tan(\alpha-15)\cot(\alpha+15) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,096 |
3.315. $\cos 2 \alpha+\sin 4 \alpha-\cos 6 \alpha$. | ## Solution.
$$
\begin{aligned}
& \cos 2 \alpha+\sin 4 \alpha-\cos 6 \alpha=(\cos 2 \alpha-\cos 6 \alpha)+\sin 4 \alpha= \\
& =\left[\cos x-\cos y=-2 \sin \frac{x+y}{2} \sin \frac{x-y}{2}\right]=-2 \sin 4 \alpha \sin (-2 \alpha)+\sin 4 \alpha= \\
& =2 \sin 4 \alpha \sin 2 \alpha+\sin 4 \alpha=2 \sin 4 \alpha\left(\sin... | 4\sin4\alpha\sin(\alpha+15)\cos(\alpha-15) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,097 |
3.316. $\sin ^{2}\left(\frac{5}{4} \pi-2 \alpha\right)-\sin ^{2}\left(\frac{5}{4} \pi+2 \alpha\right)$. | ## Solution.
$$
\begin{aligned}
& \sin ^{2}\left(\frac{5}{4} \pi-2 \alpha\right)-\sin ^{2}\left(\frac{5}{4} \pi+2 \alpha\right)=\left[\sin ^{2} \frac{x}{2}=\frac{1-\cos x}{2}\right]= \\
& =\frac{1-\cos \left(\frac{5 \pi}{2}-4 \alpha\right)}{2}-\frac{1-\cos \left(\frac{5 \pi}{2}+4 \alpha\right)}{2}= \\
& =\frac{1}{2}\l... | -\sin4\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,098 |
3.318. $\frac{3 \tan^{2}(\alpha+3 \pi)-1}{1-3 \tan^{2}\left(\alpha+\frac{5}{2} \pi\right)}$. | Solution.
$$
\begin{aligned}
& \frac{3 \tan^2(\alpha+3 \pi)-1}{1-3 \tan^2\left(\alpha+\frac{5}{2} \pi\right)}=\frac{3(\tan(3 \pi+\alpha))^2-1}{1-3\left(\tan\left(\frac{5}{2} \pi+\alpha\right)\right)^2}=\frac{3 \tan^2 \alpha-1}{1-3 \cot^2 \alpha}=\frac{3 \tan^2 \alpha-1}{1-\frac{3}{\tan^2 \alpha}}= \\
& =\frac{(3 \tan^... | \tan(\frac{\pi}{6}-\alpha)\tan(\frac{\pi}{6}+\alpha)\tan^2\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,100 |
3.320. $\frac{1-2 \sin ^{2} \alpha}{2 \operatorname{tg}\left(\frac{5}{4} \pi+\alpha\right) \cos ^{2}\left(\frac{\pi}{4}+\alpha\right)}-\operatorname{tg} \alpha+\sin \left(\frac{\pi}{2}+\alpha\right)-\cos \left(\alpha-\frac{\pi}{2}\right)$. | Solution.
$$
=\frac{1-2 \sin ^{2} \alpha}{\operatorname{tg}\left(\frac{5}{4} \pi+\alpha\right)\left(2 \cos ^{2}\left(\frac{\pi}{4}+\alpha\right)\right)}-\operatorname{tg} \alpha+\sin \left(\frac{\pi}{2}+\alpha\right)-\cos \left(\frac{\pi}{2}-\alpha\right)=
$$
$$
\begin{aligned}
& =\left[1-2 \sin ^{2} x=\cos 2 x, \ope... | \frac{2\sqrt{2}\cos(\frac{\pi}{4}+\alpha)\cos^{2}\frac{\alpha}{2}}{\cos\alpha} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,101 |
3.321. $\cos ^{2}\left(\frac{5}{8} \pi+\alpha\right)-\sin ^{2}\left(\frac{15}{8} \pi+\alpha\right)$. | Solution.
$$
\begin{aligned}
& \cos ^{2}\left(\frac{5}{8} \pi+\alpha\right)-\sin ^{2}\left(\frac{15}{8} \pi+\alpha\right)= \\
& =\left[\cos ^{2} \frac{x}{2}=\frac{1+\cos x}{2}, \sin ^{2} \frac{x}{2}=\frac{1-\cos x}{2}\right]= \\
& =\frac{1+\cos \left(\frac{5 \pi}{4}+2 \alpha\right)}{2}-\frac{1-\cos \left(\frac{15 \pi}... | \frac{\sqrt{2}}{2}\sin2\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,102 |
3.322. $\frac{2 \cos ^{2}\left(\frac{9}{4} \pi-\alpha\right)}{1+\cos \left(\frac{\pi}{2}+2 \alpha\right)}-\frac{\sin \left(\alpha+\frac{7}{4} \pi\right)}{\sin \left(\alpha+\frac{\pi}{4}\right)} \cdot \cot\left(\frac{3}{4} \pi-\alpha\right)$. | Solution.
$$
\begin{aligned}
& \frac{2 \cos ^{2}\left(\frac{9}{4} \pi-\alpha\right)}{1+\cos \left(\frac{\pi}{2}+2 \alpha\right)}-\frac{\sin \left(\alpha+\frac{7}{4} \pi\right)}{\sin \left(\alpha+\frac{\pi}{4}\right)} \cdot \operatorname{ctg}\left(\frac{3}{4} \pi-\alpha\right)= \\
& =\frac{2 \cos ^{2}\left(\frac{9}{4} ... | \frac{4\sin2\alpha}{\cos^{2}2\alpha} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,103 |
3.323. $\sin \alpha \sin ^{2}\left(\alpha-270^{\circ}\right)\left(1+\operatorname{tg}^{2} \alpha\right)+\cos \alpha \cos ^{2}\left(\alpha+270^{\circ}\right)\left(1+\operatorname{ctg}^{2} \alpha\right)$. | ## Solution.
$$
\begin{aligned}
& \sin \alpha \sin ^{2}\left(\alpha-270^{\circ}\right)\left(1+\operatorname{tg}^{2} \alpha\right)+\cos \alpha \cos ^{2}\left(\alpha+270^{\circ}\right)\left(1+\operatorname{ctg}^{2} \alpha\right)= \\
& =\sin \alpha\left(\sin \left(270^{\circ}-\alpha\right)\right)^{2}\left(1+\operatorname... | \sqrt{2}\sin(45+\alpha) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,104 |
3.324. $\sin 2 \alpha+\cos 4 \alpha-\sin 6 \alpha$. | Solution.
$\sin 2 \alpha+\cos 4 \alpha-\sin 6 \alpha=(\sin 2 \alpha-\sin 6 \alpha)+\cos 4 \alpha=$ $=\left[\sin x-\sin y=2 \cos \frac{x+y}{2} \sin \frac{x-y}{2}\right]=2 \cos 4 \alpha \sin (-2 \alpha)+\cos 4 \alpha=$ $=-2 \cos 4 \alpha \sin 2 \alpha+\cos 4 \alpha=-2 \cos 4 \alpha\left(\sin 2 \alpha-\frac{1}{2}\right)=... | 4\cos4\alpha\sin(15-\alpha)\cos(15+\alpha) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,105 |
3.325. $\cos ^{2} 2 \alpha-3 \sin ^{2} 2 \alpha$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
3.325. $\cos ^{2} 2 \alpha-3 \sin ^{2} 2 \alpha$. | Solution.
$$
\begin{aligned}
& \cos ^{2} 2 \alpha-3 \sin ^{2} 2 \alpha=\left[\cos ^{2} \frac{x}{2}=\frac{1+\cos x}{2} ; \sin ^{2} \frac{x}{2}=\frac{1-\cos x}{2}\right]= \\
& =\frac{1+\cos 4 \alpha}{2}-\frac{3(1-\cos 4 \alpha)}{2}=-1+2 \cos 4 \alpha=2\left(\cos 4 \alpha-\frac{1}{2}\right)=
\end{aligned}
$$
$=2\left(\c... | 4\sin(30+2\alpha)\sin(30-2\alpha) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,106 |
3.326. $\cos ^{2} \frac{n a}{2}-\sin ^{2} \frac{m a}{2}$.
3.326. $\cos ^{2} \frac{n a}{2}-\sin ^{2} \frac{m a}{2}$. | Solution.
$\cos ^{2} \frac{n a}{2}-\sin ^{2} \frac{m a}{2}=\left[\cos ^{2} \frac{x}{2}=\frac{1+\cos x}{2} ; \sin ^{2} \frac{x}{2}=\frac{1-\cos x}{2}\right]=$
$=\frac{1+\cos n a}{2}-\frac{1-\cos m a}{2}=\frac{1}{2}(\cos n a+\cos m a)=$
$=\left[\cos x+\cos y=2 \cos \frac{x+y}{2} \cos \frac{x-y}{2}\right]=\frac{1}{2} \... | \cos\frac{(+n)}{2}\cos\frac{(-n)}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,107 |
3.328.
$$
\frac{\cos \left(\alpha+\frac{3}{2} \pi\right)+2 \cos \left(\frac{11}{6} \pi-\alpha\right)}{2 \sin \left(\frac{\pi}{3}+\alpha\right)+\sqrt{3} \sin \left(\frac{3}{2} \pi-\alpha\right)}
$$ | ## Solution.
$$
\begin{aligned}
& \frac{\cos \left(\alpha+\frac{3}{2} \pi\right)+2 \cos \left(\frac{11}{6} \pi-\alpha\right)}{2 \sin \left(\frac{\pi}{3}+\alpha\right)+\sqrt{3} \sin \left(\frac{3}{2} \pi-\alpha\right)}=\frac{\cos \left(\frac{3}{2} \pi+\alpha\right)+2 \cos \left(2 \pi-\left(\frac{\pi}{6}+\alpha\right)\r... | \sqrt{3}\operatorname{ctg}\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,109 |
3.329. $\cos ^{2}\left(\frac{5}{4} \pi-2 \alpha\right)-\cos ^{2}\left(\frac{5}{4} \pi+2 \alpha\right)$. | Solution.
$$
\begin{aligned}
& \cos ^{2}\left(\frac{5}{4} \pi-2 \alpha\right)-\cos ^{2}\left(\frac{5}{4} \pi+2 \alpha\right)= \\
& =\frac{1+\cos \left(\frac{5 \pi}{2}-4 \alpha\right)}{2}-\frac{1+\cos \left(\frac{5 \pi}{2}+4 \alpha\right)}{2}= \\
& =\frac{1}{2}\left(\cos \left(\frac{5 \pi}{2}-4 \alpha\right)-\cos \left... | \sin4\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,110 |
3.330. $\sin \alpha-\left(\frac{\cos \left(\alpha+\frac{\pi}{4}\right)}{\cos \alpha-\sin \alpha}\right)^{2}$. | Solution.
$$
\begin{aligned}
& \sin \alpha-\left(\frac{\cos \left(\alpha+\frac{\pi}{4}\right)}{\cos \alpha-\sin \alpha}\right)^{2}=[\cos (x+y)=\cos x \cos y-\sin x \sin y]= \\
& =\sin \alpha-\left(\frac{\cos \alpha \cos \frac{\pi}{4}-\sin \alpha \sin \frac{\pi}{4}}{\cos \alpha-\sin \alpha}\right)^{2}=\sin \alpha-\left... | 2\sin(\frac{\alpha}{2}-\frac{\pi}{12})\cos(\frac{\alpha}{2}+\frac{\pi}{12}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,111 |
3.331. $\operatorname{tg} 210^{\circ}+\operatorname{ctg} 210^{\circ}+\operatorname{tg} 220^{\circ}+\operatorname{ctg} 220^{\circ}$.
3.331. $\tan 210^{\circ}+\cot 210^{\circ}+\tan 220^{\circ}+\cot 220^{\circ}$. | ## Solution.
$\operatorname{tg} 210^{\circ}+\operatorname{ctg} 210^{\circ}+\operatorname{tg} 220^{\circ}+\operatorname{ctg} 220^{\circ}=$
$=\operatorname{tg}\left(180^{\circ}+30^{\circ}\right)+\operatorname{ctg}\left(180^{\circ}+30^{\circ}\right)+\operatorname{tg}\left(180^{\circ}+40^{\circ}\right)+\operatorname{ctg}... | \frac{8}{\sqrt{3}}\sin70 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,112 |
3.332. $\frac{\sin 24^{\circ} \cos 6^{\circ}-\sin 6^{\circ} \sin 66^{\circ}}{\sin 21^{\circ} \cos 39^{\circ}-\sin 39^{\circ} \cos 21^{\circ}}=-1$. | Solution.
$$
\begin{aligned}
& \frac{\sin 24^{\circ} \cos 6^{\circ}-\sin 6^{\circ} \sin 66^{\circ}}{\sin 21^{\circ} \cos 39^{\circ}-\sin 39^{\circ} \cos 21^{\circ}}=\frac{\sin 24^{\circ} \cos 6^{\circ}-\sin 6^{\circ} \sin \left(90^{\circ}-24^{\circ}\right)}{\sin 21^{\circ} \cos 39^{\circ}-\sin 39^{\circ} \cos 21^{\cir... | -1 | Algebra | proof | Yes | Yes | olympiads | false | 48,113 |
3.333. $\frac{\sin 20^{\circ} \cos 10^{\circ}+\cos 160^{\circ} \cos 100^{\circ}}{\sin 21^{\circ} \cos 9^{\circ}+\cos 159^{\circ} \cos 99^{\circ}}=1$. | ## Solution.
$\frac{\sin 20^{\circ} \cos 10^{\circ}+\cos 160^{\circ} \cos 100^{\circ}}{\sin 21^{\circ} \cos 9^{\circ}+\cos 159^{\circ} \cos 99^{\circ}}=$
$$
\begin{aligned}
& =\frac{\sin 20^{\circ} \cos 10^{\circ}+\cos \left(180^{\circ}-20^{\circ}\right) \cos \left(90^{\circ}+10^{\circ}\right)}{\sin 21^{\circ} \cos 9... | 1 | Algebra | proof | Yes | Yes | olympiads | false | 48,114 |
3.334. $\frac{\cos 63^{\circ} \cos 3^{\circ}-\cos 87^{\circ} \cos 27^{\circ}}{\cos 132^{\circ} \cos 72^{\circ}-\cos 42^{\circ} \cos 18^{\circ}}=-\tan 24^{\circ}$. | Solution.
$$
\begin{aligned}
& \frac{\cos 63^{\circ} \cos 3^{\circ}-\cos 87^{\circ} \cos 27^{\circ}}{\cos 132^{\circ} \cos 72^{\circ}-\cos 42^{\circ} \cos 18^{\circ}}= \\
& =\frac{\cos 63^{\circ} \cos \left(90^{\circ}-87^{\circ}\right)-\cos 87^{\circ} \cos \left(90^{\circ}-63^{\circ}\right)}{\cos \left(90^{\circ}+42^{... | -\tan24 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,115 |
3.335.
$$
\frac{\cos 64^{\circ} \cos 4^{\circ}-\cos 86^{\circ} \cos 26^{\circ}}{\cos 71^{\circ} \cos 41^{\circ}-\cos 49^{\circ} \cos 19^{\circ}}=-1
$$ | Solution.
$$
\begin{aligned}
& \frac{\cos 64^{\circ} \cos 4^{\circ}-\cos 86^{\circ} \cos 26^{\circ}}{\cos 71^{\circ} \cos 41^{\circ}-\cos 49^{\circ} \cos 19^{\circ}}= \\
& =\frac{\cos \left(90^{\circ}-26^{\circ}\right) \cos 4^{\circ}-\cos \left(90^{\circ}-4^{\circ}\right) \cos 26^{\circ}}{\cos \left(90^{\circ}-19^{\ci... | -1 | Algebra | proof | Yes | Yes | olympiads | false | 48,116 |
3.336. $\frac{\cos 66^{\circ} \cos 6^{\circ}+\cos 84^{\circ} \cos 24^{\circ}}{\cos 65^{\circ} \cos 5^{\circ}+\cos 85^{\circ} \cos 25^{\circ}}=1$. | Solution.
$$
\begin{aligned}
& \frac{\cos 66^{\circ} \cos 6^{\circ}+\cos 84^{\circ} \cos 24^{\circ}}{\cos 65^{\circ} \cos 5^{\circ}+\cos 85^{\circ} \cos 25^{\circ}}= \\
& =\frac{\cos 66^{\circ} \cos 6^{\circ}+\cos \left(90^{\circ}-6^{\circ}\right) \cos \left(90^{\circ}-66^{\circ}\right)}{\cos 65^{\circ} \cos 5^{\circ}... | 1 | Algebra | proof | Yes | Yes | olympiads | false | 48,117 |
3.337. $\sin ^{2} 70^{\circ} \sin ^{2} 50^{\circ} \sin ^{2} 10^{\circ}=\frac{1}{64}$. | ## Solution.
$$
\sin ^{2} 70^{\circ} \sin ^{2} 50^{\circ} \sin ^{2} 10^{\circ}=\left(\left(\sin 70^{\circ} \sin 50^{\circ}\right) \sin 10^{\circ}\right)^{2}=
$$
$$
\begin{aligned}
& =\left[\sin x \sin y=\frac{1}{2}(\cos (x-y)-\cos (x+y))\right]= \\
& =\left(\frac{1}{2}\left(\cos 20^{\circ}-\cos 120^{\circ}\right) \si... | \frac{1}{64} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,118 |
3.338. a) $\sin 15^{\circ}=\frac{\sqrt{6}-\sqrt{2}}{4} ;$ b) $\cos 15^{\circ}=\frac{\sqrt{6}+\sqrt{2}}{4}$. | ## Solution.
a) $\sin 15^{\circ}=\sqrt{\sin ^{2} 15^{\circ}}=\sqrt{\frac{1-\cos 30^{\circ}}{2}}=\sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}}=\sqrt{\frac{2-\sqrt{3}}{4}}=$
$$
\begin{aligned}
& =\sqrt{\frac{(2-\sqrt{3}) \cdot 2}{4 \cdot 2}}=\sqrt{\frac{4-2 \sqrt{3}}{8}}=\sqrt{\frac{3-2 \sqrt{3}+1}{8}}=\sqrt{\frac{(\sqrt{3}-1)... | \sin15=\frac{\sqrt{6}-\sqrt{2}}{4};\cos15=\frac{\sqrt{6}+\sqrt{2}}{4} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,119 |
3.339. a) $\cos 36^{\circ}=\frac{\sqrt{5}+1}{4} ;$ b) $\sin 18^{\circ}=\frac{\sqrt{5}-1}{4}$. | ## Solution.
a) $\cos 36^{\circ}=\sin 54^{\circ} \Leftrightarrow \cos 2\left(18^{\circ}\right)=\sin 3\left(18^{\circ}\right)$.
## From this we get:
$$
\begin{aligned}
& 1-2 \sin ^{2} 18^{\circ}=3 \sin 18^{\circ}-4 \sin ^{3} 18^{\circ} \Leftrightarrow \\
& \Leftrightarrow 4 \sin ^{3} 18^{\circ}-2 \sin ^{2} 18^{\circ}... | \cos36=\frac{\sqrt{5}+1}{4};\sin18=\frac{\sqrt{5}-1}{4} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,120 |
3.340. $\operatorname{ctg} 10^{\circ} \operatorname{ctg} 50^{\circ} \operatorname{ctg} 70^{\circ}=\operatorname{ctg} 30^{\circ}$. | ## Solution.
$$
\begin{aligned}
& \operatorname{ctg} 10^{\circ} \operatorname{ctg} 50^{\circ} \operatorname{ctg} 70^{\circ}=\frac{\cos 10^{\circ} \cos 50^{\circ} \cos 70^{\circ}}{\sin 10^{\circ} \sin 50^{\circ} \sin 70^{\circ}}= \\
& =\left[\cos x \cos y=\frac{1}{2}(\cos (x-y)+\cos (x+y))\right.
\end{aligned}
$$
$$
\... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,121 |
3.341. $\frac{\sin 20^{\circ} \sin 40^{\circ} \sin 60^{\circ} \sin 80^{\circ}}{\sin 10^{\circ} \sin 30^{\circ} \sin 50^{\circ} \sin 70^{\circ}}=3$. | Solution.
$$
\begin{aligned}
& \frac{\left(\sin 20^{\circ} \sin 40^{\circ}\right)\left(\sin 60^{\circ} \sin 80^{\circ}\right)}{\left(\sin 10^{\circ} \sin 30^{\circ}\right)\left(\sin 50^{\circ} \sin 70^{\circ}\right)}=\left[\sin x \sin y=\frac{1}{2}(\cos (x-y)-\cos (x+y))\right]= \\
& =\frac{\left(\cos 20^{\circ}-\cos ... | 3 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,122 |
3.342. $\sin 10^{\circ} \sin 30^{\circ} \sin 50^{\circ} \sin 70^{\circ}=\frac{1}{16}$. | ## Solution.
$\left(\sin 10^{\circ} \sin 30^{\circ}\right)\left(\sin 50^{\circ} \sin 70^{\circ}\right)=$
$=\left[\sin x \sin y=\frac{1}{2}(\cos (x-y)-\cos (x+y))\right]=$
$$
=\frac{1}{2}\left(\cos 20^{\circ}-\cos 40^{\circ}\right): \frac{1}{2}\left(\cos 20^{\circ}-\cos 120^{\circ}\right)=
$$
$$
\begin{aligned}
& =\... | \frac{1}{16} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,123 |
3.343. $\sin 20^{\circ} \sin 40^{\circ} \sin 60^{\circ} \sin 80^{\circ}=\frac{3}{16}$. | ## Solution.
$\left(\sin 20^{\circ} \sin 40^{\circ}\right)\left(\sin 60^{\circ} \sin 80^{\circ}\right)=$
$$
=\left[\sin x \sin y=\frac{1}{2}(\cos (x-y)-\cos (x+y))\right]=
$$
$$
\begin{aligned}
& =\frac{1}{2}\left(\cos 20^{\circ}-\cos 60^{\circ}\right) \cdot \frac{1}{2}\left(\cos 20^{\circ}-\cos 140^{\circ}\right)= ... | \frac{3}{16} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,124 |
3.344. $\sin \frac{3 \pi}{10}-\sin \frac{\pi}{10}=\frac{1}{2}$. | Solution.
$$
\begin{aligned}
& \sin \frac{3 \pi}{10}-\sin \frac{\pi}{10}=\left[\sin 3 x=3 \sin x-4 \sin ^{3} x\right]= \\
& =3 \sin \frac{\pi}{10}-4 \sin ^{3} \frac{\pi}{10}-\sin \frac{\pi}{10}=2 \sin \frac{\pi}{10}-4 \sin ^{3} \frac{\pi}{10}= \\
& =2 \sin \frac{\pi}{10}\left(\sin ^{2} \frac{\pi}{10}+\cos ^{2} \frac{\... | \frac{1}{2} | Algebra | proof | Yes | Yes | olympiads | false | 48,125 |
3.345. $\cos \frac{\pi}{5}+\cos \frac{2 \pi}{5}+\cos \frac{4 \pi}{5}+\cos \frac{6 \pi}{5}=-\frac{1}{2}$. | ## Solution.
$$
\begin{aligned}
& \cos \frac{\pi}{5}+\cos \frac{2 \pi}{5}+\cos \frac{4 \pi}{5}+\cos \frac{6 \pi}{5}= \\
& =\cos 36^{\circ}+\cos 72^{\circ}+\cos 144^{\circ}+\cos 216^{\circ}= \\
& =\cos 36^{\circ}+\cos \left(90^{\circ}-18^{\circ}\right)+\cos \left(180^{\circ}-36^{\circ}\right)+\cos \left(270^{\circ}-54^... | -\frac{1}{2} | Algebra | proof | Yes | Yes | olympiads | false | 48,126 |
3.346. $\operatorname{ctg} 60^{\circ}+\operatorname{tg} 60^{\circ}+\operatorname{ctg} 50^{\circ}+\operatorname{tg} 50^{\circ}=\frac{8}{\sqrt{3}} \cos 20^{\circ}$. | ## Solution.
$$
\begin{aligned}
& \operatorname{ctg} 60^{\circ}+\operatorname{tg} 60^{\circ}+\operatorname{ctg} 50^{\circ}+\operatorname{tg} 50^{\circ}=\frac{\cos 60^{\circ}}{\sin 60^{\circ}}+\frac{\sin 60^{\circ}}{\cos 60^{\circ}}+\frac{\cos 50^{\circ}}{\sin 50^{\circ}}+\frac{\sin 50^{\circ}}{\cos 50^{\circ}}= \\
& =... | \frac{8}{\sqrt{3}}\cos20 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,127 |
3.347. $8 \cos \frac{4 \pi}{9} \cos \frac{2 \pi}{9} \cos \frac{\pi}{9}=1$. | ## Solution.
$$
8 \cos \frac{4 \pi}{9} \cos \frac{2 \pi}{9} \cos \frac{\pi}{9}=8 \cos 80^{\circ} \cos 40^{\circ} \cos 20^{\circ}=
$$
$=\frac{8 \cos 80^{\circ} \cos 40^{\circ} \cos 20^{\circ} \sin 20^{\circ}}{\sin 20^{\circ}}=\frac{4 \cos 80^{\circ} \cos 40^{\circ}\left(2 \cos 20^{\circ} \sin 20^{\circ}\right)}{\sin 2... | 1 | Algebra | proof | Yes | Yes | olympiads | false | 48,128 |
3.348. $\operatorname{tg} 9^{\circ}+\operatorname{tg} 15^{\circ}-\operatorname{tg} 27^{\circ}-\operatorname{ctg} 27^{\circ}+\operatorname{ctg} 9^{\circ}+\operatorname{ctg} 15^{\circ}=8$. | ## Solution.
$$
\begin{aligned}
& \operatorname{tg} 9^{\circ}+\operatorname{tg} 15^{\circ}-\operatorname{tg} 27^{\circ}-\operatorname{ctg} 27^{\circ}+\operatorname{ctg} 9^{\circ}+\operatorname{ctg} 15^{\circ}= \\
& =\left(\operatorname{tg} 9^{\circ}+\operatorname{ctg} 9^{\circ}\right)+\left(\operatorname{tg} 15^{\circ... | 8 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,129 |
3.349. $\frac{\sin \left(\alpha-\frac{3}{2} \pi\right) \tan\left(\frac{\pi}{4}+\frac{\alpha}{2}\right)}{1+\cos \left(\alpha-\frac{5}{2} \pi\right)}=1$. | Solution.
$$
\begin{aligned}
& \frac{\sin \left(\alpha-\frac{3}{2} \pi\right) \tan\left(\frac{\pi}{4}+\frac{\alpha}{2}\right)}{1+\cos \left(\alpha-\frac{5}{2} \pi\right)}=\frac{-\sin \left(\frac{3}{2} \pi-\alpha\right) \tan\left(\frac{\pi}{4}+\frac{\alpha}{2}\right)}{1+\cos \left(\frac{5}{2} \pi-\alpha\right)}= \\
& =... | 1 | Algebra | proof | Yes | Yes | olympiads | false | 48,130 |
3.350. $\cos 70^{\circ}+8 \cos 20^{\circ} \cos 40^{\circ} \cos 80^{\circ}=2 \cos ^{2} 35^{\circ}$. | ## Solution.
$\cos 70^{\circ}+8 \cos 20^{\circ} \cos 40^{\circ} \cos 80^{\circ}=$
$$
\begin{aligned}
& =\cos 2\left(35^{\circ}\right)+\frac{8 \cos 20^{\circ} \cos 40^{\circ} \cos 80^{\circ} \cdot \sin 20^{\circ}}{\sin 20^{\circ}}= \\
& =\cos 2\left(35^{\circ}\right)+\frac{4\left(2 \sin 20^{\circ} \cos 20^{\circ}\righ... | 2\cos^{2}35 | Algebra | proof | Yes | Yes | olympiads | false | 48,131 |
3.351. $1-\cos \left(\frac{3}{2} \pi-3 \alpha\right)-\sin ^{2} \frac{3}{2} \alpha+\cos ^{2} \frac{3}{2} \alpha=2 \sqrt{2} \cos \frac{3}{2} \alpha \sin \left(\frac{3 \alpha}{2}+\frac{\pi}{4}\right)$. | ## Solution.
$$
\begin{aligned}
& 1-\cos \left(\frac{3}{2} \pi-3 \alpha\right)-\sin ^{2} \frac{3}{2} \alpha+\cos ^{2} \frac{3}{2} \alpha= \\
& =1+\sin 3 \alpha-\sin ^{2} \frac{3 \alpha}{2}+\cos ^{2} \frac{3 \alpha}{2}=\sin 3 \alpha+2 \cos ^{2} \frac{3 \alpha}{2}= \\
& =\sin 2\left(\frac{3 \alpha}{2}\right)+2 \cos ^{2}... | proof | Algebra | proof | Yes | Yes | olympiads | false | 48,132 |
3.352.
$$
\frac{\cos \left(2 \alpha-\frac{\pi}{2}\right)+\sin (3 \pi-4 \alpha)-\cos \left(\frac{5}{2} \pi+6 \alpha\right)}{4 \sin (5 \pi-3 \alpha) \cos (\alpha-2 \pi)}=\cos 2 \alpha
$$ | Solution.
$$
\begin{aligned}
& \frac{\cos \left(2 \alpha-\frac{\pi}{2}\right)+\sin (3 \pi-4 \alpha)-\cos \left(\frac{5}{2} \pi+6 \alpha\right)}{4 \sin (5 \pi-3 \alpha) \cos (\alpha-2 \pi)}= \\
& =\frac{\cos \left(\frac{\pi}{2}-2 \alpha\right)+\sin (3 \pi-4 \alpha)-\cos \left(\frac{5}{2} \pi+6 \alpha\right)}{4 \sin (5 ... | \cos2\alpha | Algebra | proof | Yes | Yes | olympiads | false | 48,133 |
3.353. $\frac{1}{\sin 10^{\circ}}-\frac{\sqrt{3}}{\cos 10^{\circ}}=4$. | Solution.
$$
\begin{aligned}
& \frac{1}{\sin 10^{\circ}}-\frac{\sqrt{3}}{\cos 10^{\circ}}=\frac{\cos 10^{\circ}-\sqrt{3} \sin 10^{\circ}}{\sin 10^{\circ} \cos 10^{\circ}}=\frac{2\left(\frac{1}{2} \cos 10^{\circ}-\frac{\sqrt{3}}{2} \sin 10^{\circ}\right)}{\sin 10^{\circ} \cos 10^{\circ}}= \\
& =\frac{2 \cdot 2\left(\si... | 4 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,134 |
3.354. $\cos 36^{\circ}-\sin 18^{\circ}=\sin 30^{\circ}$. | Solution.
$\cos 36^{\circ}-\sin 18^{\circ}=$
$=\left[\cos 36^{\circ}=\frac{\sqrt{5}+1}{4} ; \sin 18^{\circ}=\frac{\sqrt{5}-1}{4}\right.$ (see № 3.339 a) and b)) $]=$
$=\frac{\sqrt{5}+1}{4}-\frac{\sqrt{5}-1}{4}=\frac{2}{4}=\frac{1}{2}=\sin 30^{\circ}$.
The equality is valid.
Calculate: (3.355-3.367): | \sin30=\frac{1}{2} | Algebra | proof | Yes | Yes | olympiads | false | 48,135 |
3.355. $\sin ^{4} \frac{\pi}{8}+\cos ^{4} \frac{3 \pi}{8}+\sin ^{4} \frac{5 \pi}{8}+\cos ^{4} \frac{7 \pi}{8}$. | ## Solution.
$$
\begin{aligned}
& \sin ^{4} \frac{\pi}{8}+\cos ^{4} \frac{3 \pi}{8}+\sin ^{4} \frac{5 \pi}{8}+\cos ^{4} \frac{7 \pi}{8}= \\
& =\left(\sin ^{2} \frac{\pi}{8}\right)^{2}+\left(\cos ^{2} \frac{3 \pi}{8}\right)^{2}+\left(\sin ^{2} \frac{5 \pi}{8}\right)^{2}+\left(\cos ^{2} \frac{7 \pi}{8}\right)^{2}= \\
& ... | \frac{3}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,136 |
3.356. $\sin 20^{\circ} \cos 50^{\circ} \sin 60^{\circ} \cos 10^{\circ}$. | Solution.
$\sin 20^{\circ} \cos 50^{\circ} \sin 60^{\circ} \cos 10^{\circ}=\left(\sin 20^{\circ} \cos 10^{\circ}\right)\left(\sin 60^{\circ} \cos 50^{\circ}\right)=$
$=\left[\sin x \cos y=\frac{1}{2}(\sin (x-y)+\sin (x+y))\right]=$
$=\frac{1}{2}\left(\sin 10^{\circ}+\sin 30^{\circ}\right) \cdot \frac{1}{2}\left(\sin... | \frac{3}{16} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,137 |
3.357. $\cos \frac{3 \pi}{5} \cos \frac{6 \pi}{5}$. | Solution.
$$
\begin{aligned}
& \cos \frac{3 \pi}{5} \cos \frac{6 \pi}{5}=\cos \frac{5 \pi-2 \pi}{5} \cos \frac{5 \pi+\pi}{5}=\cos \left(\pi-\frac{2 \pi}{5}\right) \cos \left(\pi+\frac{\pi}{5}\right)= \\
& =-\cos \frac{2 \pi}{5} \cdot\left(-\cos \frac{\pi}{5}\right)=\cos \frac{2 \pi}{5} \cos \frac{\pi}{5}=\left(2 \cos ... | \frac{1}{4} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,138 |
3.358. $\frac{\cos 68^{\circ} \cos 8^{\circ}-\cos 82^{\circ} \cos 22^{\circ}}{\cos 53^{\circ} \cos 23^{\circ}-\cos 67^{\circ} \cos 37^{\circ}}$. | ## Solution.
$$
\begin{aligned}
& \frac{\cos 68^{\circ} \cos 8^{\circ}-\cos 82^{\circ} \cos 22^{\circ}}{\cos 53^{\circ} \cos 23^{\circ}-\cos 67^{\circ} \cos 37^{\circ}}= \\
& =\frac{\cos 68^{\circ} \cos 8^{\circ}-\cos \left(90^{\circ}-8^{\circ}\right) \cos \left(90^{\circ}-68^{\circ}\right)}{\cos 53^{\circ} \cos 23^{\... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,139 |
3.359. $\frac{\cos 70^{\circ} \cos 10^{\circ}+\cos 80^{\circ} \cos 20^{\circ}}{\cos 69^{\circ} \cos 9^{\circ}+\cos 81^{\circ} \cos 21^{\circ}}$. | ## Solution.
$$
\frac{\cos 70^{\circ} \cos 10^{\circ}+\cos 80^{\circ} \cos 20^{\circ}}{\cos 69^{\circ} \cos 9^{\circ}+\cos 81^{\circ} \cos 21^{\circ}}=
$$
$$
\begin{aligned}
& =\frac{\cos \left(90^{\circ}-20^{\circ}\right) \cos 10^{\circ}+\cos \left(90^{\circ}-10^{\circ}\right) \cos 20^{\circ}}{\cos \left(90^{\circ}-... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,140 |
3.360. $\frac{\cos 67^{\circ} \cos 7^{\circ}-\cos 83^{\circ} \cos 23^{\circ}}{\cos 128^{\circ} \cos 68^{\circ}-\cos 38^{\circ} \cos 22^{\circ}}-\operatorname{tg} 164^{\circ}$.
| ## Решение.
$$
\frac{\cos 67^{\circ} \cos 7^{\circ}-\cos 83^{\circ} \cos 23^{\circ}}{\cos 128^{\circ} \cos 68^{\circ}-\cos 38^{\circ} \cos 22^{\circ}}-\operatorname{tg} 164^{\circ}=
$$
$$
=\frac{\cos 67^{\circ} \cos \left(90^{\circ}-83^{\circ}\right)-\cos 83^{\circ} \cos \left(90^{\circ}-67^{\circ}\right)}{\cos \left... | 0 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,141 |
3.361. $\frac{\sin 22^{\circ} \cos 8^{\circ}+\cos 158^{\circ} \cos 98^{\circ}}{\sin 23^{\circ} \cos 7^{\circ}+\cos 157^{\circ} \cos 97^{\circ}}$. | Solution.
$$
\begin{aligned}
& \frac{\sin 22^{\circ} \cos 8^{\circ}+\cos 158^{\circ} \cos 98^{\circ}}{\sin 23^{\circ} \cos 7^{\circ}+\cos 157^{\circ} \cos 97^{\circ}}= \\
& =\frac{\sin 22^{\circ} \cos 8^{\circ}+\cos \left(180^{\circ}-22^{\circ}\right) \cos \left(90^{\circ}+8^{\circ}\right)}{\sin 23^{\circ} \cos 7^{\ci... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 48,142 |
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