Question

The value of $\cos 0+\cos \frac{\pi }{7}+\cos \frac{2\pi }{7}+\cos \frac{3\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{5\pi }{7}+\cos \frac{6\pi }{7}=$

Answer 1

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

$1+[\cos \frac{\pi }{7}+\cos \frac{6\pi }{7}]+[\cos \frac{2\pi }{7}+\cos \frac{5\pi }{7}]+[\cos \frac{3\pi }{7}+\cos \frac{4\pi }{7}]$

$=1+2\cos \frac{\pi }{2}.\cos \frac{5\pi }{14}+2\cos \frac{\pi }{2}.\cos \frac{3\pi }{14}+2\cos \frac{\pi }{2}.\cos \frac{\pi }{14}$

$=1+0+0+0=1$