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

Answer: (B)

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