Question

The value of $\cos \frac{\pi}{5} \cos \frac{2\pi}{5} \cos \frac{4\pi}{5} \cos \frac{8\pi}{5} $ is

• A. $\frac{1}{16}$
• B. $ - \frac{1}{16}$
• C. 1
• D. 0

Answer 1

Answer: (B)

$\cos \frac{\pi}{5} \cos \frac{2\pi}{5} \cos \frac{4\pi}{5} \cos \frac{8\pi}{5} $
$=\cos\theta \cos 2\theta \cos2^{2} \theta \cos2^{3} \theta ,$ where $ \theta =\frac{ \pi}{5}$
$ = \frac{\sin2^{4} \theta}{2^{4} \sin\theta} = \frac{\sin16\theta}{16 \sin\theta } = \frac{\sin\left(15\theta + \theta\right)}{16 \sin\theta} $
$= \frac{\sin\left(3 \pi+\theta\right)}{16 \sin\theta} = \frac{-\sin\theta}{16 \sin \theta} = - \frac{1}{16} $