# The value of cos

Question:

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

Solution:

$\cos \frac{\pi}{5} \cos \frac{2 \pi}{5} \cos \frac{4 \pi}{5} \cos \frac{8 \pi}{5}$

multiply and divide above equation by $2 \sin \frac{\pi}{5}$

$=\frac{1}{2 \sin \frac{\pi}{5}}\left(2 \sin \frac{\pi}{5} \cos \frac{\pi}{5} \cos \frac{2 \pi}{5} \cos \frac{4 \pi}{5} \cos \frac{8 \pi}{5}\right)$

$=\frac{1}{2 \sin \frac{\pi}{5}}\left(\sin \frac{2 \pi}{5} \cos \frac{2 \pi}{5} \cos \frac{4 \pi}{5} \cos \frac{8 \pi}{5}\right) \quad($ using identity $2 \sin \theta \cos \theta=\sin 2 \theta)$

$=\frac{1}{2^{2} \sin \frac{\pi}{5}}\left(2 \sin \frac{2 \pi}{5} \cos \frac{2 \pi}{5} \cos \frac{4 \pi}{5} \cos \frac{8 \pi}{5}\right)$

$=\frac{1}{2^{2} \sin \frac{\pi}{5}}\left(\sin \frac{4 \pi}{5} \cos \frac{4 \pi}{5} \cos \frac{8 \pi}{5}\right)$

$=\frac{1}{2^{3} \sin \frac{\pi}{5}}\left(2 \sin \frac{4 \pi}{5} \cos \frac{4 \pi}{5} \cos \frac{8 \pi}{5}\right)$

$=\frac{1}{2^{3} \sin \frac{\pi}{5}}\left(\sin \frac{8 \pi}{5} \cos \frac{8 \pi}{5}\right)$

$=\frac{1}{2^{4} \sin \frac{\pi}{5}}\left(2 \sin \frac{8 \pi}{5} \cos \frac{8 \pi}{5}\right)$

$=\frac{1}{2^{4} \sin \frac{\pi}{5}}\left(\sin \frac{16 \pi}{5}\right)$

$=\frac{1}{16 \sin \frac{\pi}{5}}(\sin (3 \pi+\pi / 5))$

$=\frac{1}{16 \sin \frac{\pi}{5}}\left(\sin \frac{\pi}{5}\right) \quad(\because \sin (3 \pi+\theta)=-\sin \theta)$

$\therefore \cos \frac{\pi}{5} \cos \frac{2 \pi}{5} \cos \frac{4 \pi}{5} \cos \frac{8 \pi}{5}=-\frac{1}{16}$