Question

The value of $\cos \frac{2\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{6\pi }{7}$ is

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

Answer 1

The correct option is D $-\frac{1}{2}$

$\cos \frac{2\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{6\pi }{7}=$

Multiply and divide by $\sin \frac{\pi }{7}$,we get

$=\frac{\sin \frac{\pi }{7}.\cos \frac{2\pi }{7}+\sin \frac{\pi }{7}.\cos \frac{4\pi }{7}+\sin \frac{\pi }{7}\cos \frac{6\pi }{7}}{\sin \frac{\pi }{7}}$

Convert product form to sum form $2\sin A\cos B=\sin (A-B)+\cos (A+B)$

$=\frac{\sin (\frac{\pi }{7}-\frac{2\pi }{7})+\sin (\frac{\pi }{7}+\frac{2\pi }{7})+\sin (\frac{\pi }{7}-\frac{4\pi }{7})+\sin (\frac{\pi }{7}+\frac{4\pi }{7})+\sin (\frac{\pi }{7}-\frac{6\pi }{7})+\sin (\frac{\pi }{7}+\frac{6\pi }{7})}{2\sin \frac{\pi }{7}}$

$=\frac{-\sin \frac{\pi }{7}+\sin \frac{3\pi }{7}-\sin \frac{3\pi }{7}+\sin \frac{5\pi }{7}-\sin \frac{5\pi }{7}+\sin \frac{7\pi }{7}}{\sin \frac{\pi }{7}}$

$=\frac{-\sin \frac{\pi }{7}+\sin \pi }{2\sin \frac{\pi }{7}}$

$=\frac{-\sin \frac{\pi }{7}}{2\sin \frac{\pi }{7}}$

$=-\frac{1}{2}$