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] $

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