Question

Simplify the following expression:
$\cos 0+\cos \frac{\pi }{2}+\cos \frac{2\pi }{7}+\cos \frac{3\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{5\pi }{7}+\cos \frac{6\pi }{7}$

Answer 1

$\cos 0°+\cos \frac{\pi }{2}+\cos \frac{2\pi }{7}+\cos \frac{3\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{5\pi }{7}+\cos \frac{6\pi }{7}$

$=1+0+\cos \frac{6\pi }{7}+2\cos \frac{\pi }{2}\cos \frac{3\pi }{14}+2\cos \frac{\pi }{2}\cos \frac{\pi }{14}$

$=1+\cos \frac{6\pi }{7}={2\cos }^2\frac{3\pi }{7}$