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)

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