Question

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

• A. 2
• B. $- \frac{1}{2}$
• C. $\frac{1}{2}$
• D. none of these.

Answer 1

Answer: (B)

$- \frac{1}{2}$

Answer 2

$\cos \frac{2\pi}{7} + \cos \frac{4 \pi}{7} + \cos \frac{6\pi}{7}$ = \frac{1}{2 \sin \frac{\pi}{7}}\left\\{ 2 \cos \frac{2\pi}{7} \sin \frac{\pi}{7} + 2 \cos \frac{4\pi}{7} \sin \frac{\pi}{7} + 2 \cos \frac{6\pi}{7} \sin \frac{\pi}{7} \right\\} = \frac{1}{2 \sin \frac{\pi}{7}} \left\\{\sin \frac{3\pi}{7} -\sin \frac{\pi}{7} + \sin \frac{5 \pi}{7} - \sin \frac{3\pi}{7} + \sin \frac{7\pi}{7} - \sin \frac{5\pi}{7}\right\\} $= - \frac{1}{2} \left(\because \sin \pi= 0 \right)$