Find the Value of cos(π/7) cos(2π/7) cos(4π/7)
Question
Find the value of
\[ \cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{4\pi}{7}. \]Solution
Use the standard identity
\[ \cos x \cos 2x \cos 4x = \frac{\sin 8x}{8\sin x}. \]Putting
\[ x=\frac{\pi}{7}, \]we get
\[ \cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{4\pi}{7} = \frac{\sin\frac{8\pi}{7}} {8\sin\frac{\pi}{7}}. \]Since
\[ \sin\frac{8\pi}{7} = \sin\left(\pi+\frac{\pi}{7}\right) = -\sin\frac{\pi}{7}, \]therefore
\[ \cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{4\pi}{7} = \frac{-\sin\frac{\pi}{7}} {8\sin\frac{\pi}{7}} = -\frac{1}{8}. \]