Question

The value of $\cos \frac{\pi }{7}+\cos \frac{2\pi }{7}+\cos \frac{3\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{5\pi }{7}+\cos \frac{6\pi }{7}+\cos \frac{7\pi }{7}$ is

• A. $0$
• B. $-1$
• C. $1$
• D. None of these

Answer 1

The correct option is B $-1$

We have,
$\cos \frac{\pi }{7}+\cos \frac{2\pi }{7}+\cos \frac{3\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{5\pi }{7}+\cos \frac{6\pi }{7}+\cos \frac{7\pi }{7}$
$=(\cos \frac{\pi }{7}+\cos \frac{6\pi }{7})+(\cos \frac{2\pi }{7}+\cos \frac{5\pi }{7})+(\cos \frac{3\pi }{7}+\cos \frac{4\pi }{7})+\cos \pi$
$=(\cos \frac{\pi }{7}+\cos (\pi -\frac{\pi }{7}))+(\cos \frac{2\pi }{7}+\cos (\pi -\frac{2\pi }{7}))+(\cos \frac{3\pi }{7}+\cos (\pi -\frac{3\pi }{7}))+(-1)$
$=0+0+0-1=-1$