Question

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

• A. $0$
• B. $1$
• C. $-1$
• D. $2$

Answer 1

The correct option is C $-1$

Let $\cos \frac{\pi }{7}+\cos \frac{2\pi }{7}+\cos \frac{3\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{5\pi }{7}+\cos \frac{6\pi }{7}+\cos \frac{7\pi }{7}=I$

$I=\cos \frac{\pi }{7}+\cos \frac{6\pi }{7}+\cos \frac{2\pi }{7}+\cos \frac{5\pi }{7}+\cos \frac{3\pi }{7}+\cos \frac{4\pi }{7}+\cos \pi$

Using $\cos A+\cos B=2\cos \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)$

$I=2\cos \frac{\pi }{2}\cos (-\frac{5\pi }{14})+2\cos \frac{\pi }{2}\cos (-\frac{3\pi }{14})+2\cos \frac{\pi }{2}\cos (-\frac{\pi }{14})-1$

$=0+0+0-1$