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)

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