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. $1$
• B. $-1$
• C. $0$
• D. none of these

Answer 1

Answer: (B)

$-1$

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

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

$\Rightarrow a=(\cos \frac{\pi }{7}+\cos \frac{6\pi }{7})+(\cos \frac{2\pi }{7}+\cos \frac{5\pi }{7})+(\cos \frac{3\pi }{7}+\cos \frac{4\pi }{7})+\cos \pi$

$\Rightarrow a=2\cos \left[\frac{1}{2}(\frac{\pi }{7}+\frac{6\pi }{7})\right]\cos \left[\frac{1}{2}(\frac{\pi }{7}-\frac{6\pi }{7})\right]+2\cos \left[\frac{1}{2}(\frac{2\pi }{7}+\frac{5\pi }{7})\right]\cos \left[\frac{1}{2}(\frac{2\pi }{7}-\frac{5\pi }{7})\right]+2\cos \left[\frac{1}{2}(\frac{3\pi }{7}+\frac{4\pi }{7})\right]\cos \left[\frac{1}{2}(\frac{3\pi }{7}-\frac{4\pi }{7})\right]-1$

$\Rightarrow a=2\cos \frac{\pi }{2}[\cos \frac{5\pi }{14}+\cos \frac{3\pi }{14}+\cos \frac{\pi }{14}]-1$

$\therefore a=-1$

Ans-Option $B$.