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Question
Write the value of \[\cos\frac{\pi}{7} \cos\frac{2\pi}{7} \cos\frac{4\pi}{7} .\]
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Solution
\[\text{ We have } , \]
\[ \cos\frac{\pi}{7} \cos\frac{2\pi}{7} \cos\frac{4\pi}{7} = \frac{2\sin\frac{\pi}{7} \cos\frac{\pi}{7} \cos\frac{2\pi}{7} \cos\frac{4\pi}{7}}{2\sin\frac{\pi}{7}} \]
\[ \left[ \text{ On dividing and multiplying by } 2\sin\frac{\pi}{7} \right]\]
\[ = \frac{2 \times \sin\frac{2\pi}{7} \cos\frac{2\pi}{7} \cos\frac{4\pi}{7}}{2 \times 2\sin\frac{\pi}{7}}\]
Proceeding in the same way, we get
\[\cos\frac{\pi}{7} \cos\frac{2\pi}{7} \cos\frac{4\pi}{7} = \frac{\sin\frac{8\pi}{7}}{8\sin\frac{\pi}{7}}\]
\[ = \frac{\sin\left( \pi + \frac{\pi}{7} \right)}{8\sin\frac{\pi}{7}}\]
\[ = \frac{- \sin\frac{\pi}{7}}{8\sin\frac{\pi}{7}}\]
\[ \therefore \cos\frac{\pi}{7} \cos\frac{2\pi}{7} \cos\frac{4\pi}{7} = \frac{- 1}{8}\]
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