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Question
Prove that: \[\cos\frac{\pi}{5}\cos\frac{2\pi}{5}\cos\frac{4\pi}{5}\cos\frac{8\pi}{5} = \frac{- 1}{16}\]
Solution
\[\cos\frac{\pi}{5}\cos\frac{2\pi}{5}\cos\frac{4\pi}{5}\cos\frac{8\pi}{5}\]
\[ = \frac{1}{2\sin\frac{\pi}{5}}\left( 2\sin\frac{\pi}{5}\cos\frac{\pi}{5}\cos\frac{2\pi}{5}\cos\frac{4\pi}{5}\cos\frac{8\pi}{5} \right) \left( \text{ Multiplying and dividing by } \frac{1}{2\sin\frac{\pi}{5}} \right)\]
\[ = \frac{1}{2\sin\frac{\pi}{5}}\left( \sin\frac{2\pi}{5}\cos\frac{2\pi}{5}\cos\frac{4\pi}{5}\cos\frac{8\pi}{5} \right) \left( \sin2A = 2\text{ sin } A\text{ cos } A \right)\]
\[ = \frac{1}{4\sin\frac{\pi}{5}}\left( 2\sin\frac{2\pi}{5}\cos\frac{2\pi}{5}\cos\frac{4\pi}{5}\cos\frac{8\pi}{5} \right) \left( \text{ Multiplying and dividing by } 2 \right)\]
\[= \frac{1}{4\sin\frac{\pi}{5}}\left( \sin\frac{4\pi}{5}\cos\frac{4\pi}{5}\cos\frac{8\pi}{5} \right)\]
\[ = \frac{1}{8\sin\frac{\pi}{5}}\left( 2\sin\frac{4\pi}{5}\cos\frac{4\pi}{5}\cos\frac{8\pi}{5} \right) \left( \text{ Multiplying and dividing by 2 } \right)\]
\[ = \frac{1}{8\sin\frac{\pi}{5}}\left( \sin\frac{8\pi}{5}\cos\frac{8\pi}{5} \right)\]
\[ = \frac{1}{16\sin\frac{\pi}{5}}\left( 2\sin\frac{8\pi}{5}\cos\frac{8\pi}{5} \right) \left(\text{ Multiplying and dividing by 2 } \right)\]
\[= \frac{\sin\frac{16\pi}{5}}{16\sin\frac{\pi}{5}}\]
\[ = \frac{\sin\left( 3\pi + \frac{\pi}{5} \right)}{16\sin\frac{\pi}{5}}\]
\[ = \frac{- \sin\frac{\pi}{5}}{16\sin\frac{\pi}{5}} \left[ \sin\left( 3\pi + \theta \right) = - \sin\theta \right]\]
\[ = \frac{- 1}{16}\]
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