Question

The value of $\cos \frac{\pi }{15}\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}$ is

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

Answer 1

Answer: (B)

$\cos \frac{\pi }{15}\cdot \cos \frac{2\pi }{15}\cdot \cos \frac{4\pi }{15}\cdot \cos \frac{8\pi }{15}$

$=\frac{1}{4}\left(2\cos \frac{4\pi }{15}\cos \frac{\pi }{15}\right)\left(2\cos \frac{8\pi }{15}\cos \frac{2\pi }{15}\right)$

$=\frac{1}{4}\left(\cos 60^{\circ }+\cos 36^{\circ }\right)\left(\cos 120^{\circ }+\cos 72^{\circ }\right)$

$=\frac{1}{4}\left(\frac{1}{2}+\frac{\sqrt{5}+1}{4}\right)\left(-\frac{1}{2}+\frac{\sqrt{5}-1}{4}\right)$

$\left(\because \cos 36^{\circ }=\frac{\sqrt{5}+1}{4}\text{and}\cos 72^{\circ }=\frac{\sqrt{5}-1}{4}\right)$

$=\frac{1}{4}\left[-\frac{1}{4}+\frac{1}{2}\left(\frac{\sqrt{5}-1}{4}-\frac{\sqrt{5}+1}{4}\right)+\frac{5-1}{16}\right]$

$=\frac{1}{4}\left[\frac{1}{2}\left(-\frac{1}{2}\right)\right]=-\frac{1}{16}$