Question

$\cos \frac{\pi }{7}\cos \frac{2\pi }{7}\cos \frac{4\pi }{7}=$

• A. $0$
• B. $\frac{1}{2}$
• C. $\frac{1}{4}$
• D. $-\frac{1}{8}$

Answer 1

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

$\cos \frac{\pi }{7}\cos \frac{2\pi }{7}\cos \frac{4\pi }{7}$

$=\frac{1}{2\sin \frac{\pi }{7}}\left(2\sin \frac{\pi }{7}\cos \frac{\pi }{7}\right)\cos \frac{2\pi }{7}\cos \frac{4\pi }{7}$

$=\frac{1}{2\sin \frac{\pi }{7}}\left(\sin \frac{2\pi }{7}\cos \frac{2\pi }{7}\right)\cos \frac{4\pi }{7}$

$=\frac{1}{2^2\sin \frac{\pi }{7}}\sin \frac{4\pi }{7}\cos \frac{4\pi }{7}$

$=\frac{1}{2^3\sin \frac{\pi }{7}}\sin \frac{8\pi }{7}$

$=\frac{1}{8}\frac{\sin (\pi +\frac{\pi }{7})}{\sin \frac{\pi }{7}}=\frac{1}{8}\frac{-\sin \left(\frac{\pi }{7}\right)}{\sin \frac{\pi }{7}}=-\frac{1}{8}$