Question

The value of $ sin \frac{\pi}{16}\, sin \frac{3\pi}{16} \sin \frac{5\pi}{16} \sin \frac{7\pi}{16} $ is :

• A. $ \frac{\sqrt{2}}{16} $
• B. $ \frac{1}{8} $
• C. $ \frac{1}{16} $
• D. $ \frac{\sqrt{2}}{32} $

Answer 1

Answer: (A)

$sin \frac{\pi}{16}\cdot sin \frac{3\pi}{16} \cdot sin \frac{5\pi}{16} \cdot sin \frac{7\pi}{16} $
$ = \frac{1}{2}\left[2 sin \frac{5\pi}{16} sin \frac{3\pi}{16}\right]\times\frac{1}{2}\left[2 sin \frac{7\pi}{16} sin \frac{\pi}{16}\right] $
$ = \frac{1}{4}\left[\left(cos \frac{\pi}{8}-cos \frac{\pi}{2}\right)\left(cos \frac{3\pi}{8} - cos \frac{\pi}{2}\right)\right] $
$= \frac{1}{4\times2} \left(cos \frac{\pi}{2} + cos \frac{\pi}{4}\right) $
$= \frac{1}{8}\left(\frac{1}{\sqrt{2}}\right) $
$= \frac{1}{8\sqrt{2}} $
$= \frac{\sqrt{2}}{16} \left[ \because cos \frac{\pi}{2} = 0\right]$