Question

The value of $\cos \frac{\pi }{7}\cos \frac{2\pi }{7}\cos \frac{3\pi }{7}$ is

• A. 0
• B. $\frac{1}{2}$
• C. $\frac{1}{3}$
• D. $\frac{1}{8}$

Answer 1

The correct option is D $\frac{1}{8}$

The value of $\cos \frac{\pi }{7}\cos \frac{2\pi }{7}\cos \frac{3\pi }{7}$
$=\frac{2\sin \frac{\pi }{7}\cos \frac{\pi }{7}\cos \frac{2\pi }{7}\cos \frac{3\pi }{7}}{2\sin \frac{\pi }{7}}$
$=\frac{\sin \frac{2\pi }{7}\cos \frac{2\pi }{7}\cos \frac{3\pi }{7}}{2\sin \frac{\pi }{7}}$
$=\frac{\sin \frac{4\pi }{7}\cos \frac{3\pi }{7}}{4\sin \frac{\pi }{7}}=\frac{\sin \frac{3\pi }{7}\cos \frac{3\pi }{7}}{4\sin \frac{\pi }{7}}$
$=\frac{\sin \frac{6\pi }{7}}{8\sin \frac{\pi }{7}}=\frac{\sin (\pi -\frac{\pi }{7})}{8\sin \frac{\pi }{7}}=\frac{\sin \frac{\pi }{7}}{8\sin \frac{\pi }{7}}=\frac{1}{8}$