Question

$sin\frac{\pi }{14}sin\frac{3\pi }{14}sin\frac{5\pi }{14}sin\frac{7\pi }{14}$ is equal to

• A. 1
• B. $\frac{1}{4}$
• C. $\frac{1}{8}$
• D. $\frac{\sqrt{2}}{7}$

Answer 1

Answer: (C)

$\sin \frac{\pi }{14}\sin \frac{3\pi }{14}\sin \frac{5\pi }{14}\sin \frac{7\pi }{14}$
$=\sin \left(\frac{\pi }{2}-\frac{3\pi }{7}\right)\sin \left(\frac{\pi }{2}-\frac{2\pi }{7}\right)\sin \left(\frac{\pi }{2}-\frac{\pi }{7}\right)\sin \frac{\pi }{2}$
$=\cos \frac{3\pi }{7}\cos \frac{2\pi }{7}\cos \frac{\pi }{7}$
$=-\cos \frac{\pi }{7}\cos \frac{2\pi }{7}\cos \frac{4\pi }{7}$
$=-\frac{\sin \left(2^3\pi /7\right)}{2^3\sin \pi /7}=-\frac{\sin 8\pi /7}{8\sin \pi /7}=\frac{1}{8}$