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Question
The value of \[\cos \frac{\pi}{65} \cos \frac{2\pi}{65} \cos \frac{4\pi}{65} \cos \frac{8\pi}{65} \cos \frac{16\pi}{65} \cos \frac{32\pi}{65}\] is
Options
- \[\frac{1}{8}\]
- \[\frac{1}{16}\]
- \[\frac{1}{32}\]
none of these
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Solution
none of these
\[\text{ We have } , \]
\[\cos\frac{\pi}{65} \cos\frac{2\pi}{65} \cos\frac{4\pi}{65} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65}\]
\[ = \frac{2\sin\frac{\pi}{65}}{2\sin\frac{\pi}{65}} \cos\frac{\pi}{65} \cos\frac{2\pi}{65} \cos\frac{4\pi}{65} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65}\]
\[ \left( \text{ dividing and multiplying by } 2\sin\frac{\pi}{65} \right)\]
\[ = \frac{2\sin\frac{2\pi}{65}}{2 \times 2\sin\frac{\pi}{65}} \cos\frac{2\pi}{65} \cos\frac{4\pi}{65} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65}\]
\[ = \frac{2\sin\frac{4\pi}{65}}{2 \times 4\sin\frac{\pi}{65}} \cos\frac{4\pi}{65} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65}\]
\[= \frac{2\sin\frac{8\pi}{65}}{2 \times 8\sin\frac{\pi}{65}} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65}\]
\[ = \frac{2\sin\frac{16\pi}{65}}{2 \times 16\sin\frac{\pi}{65}} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65}\]
\[ = \frac{2\sin\frac{32\pi}{65}}{2 \times 32\sin\frac{\pi}{65}} \cos\frac{32\pi}{65}\]
\[ = \frac{\sin\frac{64\pi}{65}}{64\sin\frac{\pi}{65}}\]
\[ = \frac{\sin\left( \pi - \frac{\pi}{65} \right)}{64\sin\frac{\pi}{65}}\]
\[ = \frac{\sin\frac{\pi}{65}}{64\sin\frac{\pi}{65}}\]
\[ = \frac{1}{64}\]
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