Question

Find : $\cos \frac{2\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{6\pi }{7}+\frac{3}{2}$

Answer 1

$\cos \frac{2\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{6\pi }{7}+\frac{1}{2}$

to multiply by $\sin \frac{\pi }{7}$

$\sin \frac{\pi }{7}\cos \frac{2\pi }{7}+\sin \frac{\pi }{7}\cos 4\frac{\pi }{7}+\sin \frac{\pi }{7}\cos \frac{6\pi }{7}+\frac{1}{2}\sin \frac{\pi }{7}$

$\Rightarrow \sin A\cos B=\frac{1}{2}\left[\sin \right(A-B)+\sin (A+B\left)\right]$

$\Rightarrow \frac{1}{2}[\sin (\frac{\pi }{2}-\frac{2\pi }{7})+\sin (\frac{\pi }{7}+\frac{2\pi }{7})]$

$+\frac{1}{2}[\sin (\frac{\pi }{7}-\frac{4\pi }{7})+\sin (\frac{\pi }{7}+\frac{4\pi }{7})]$

$+\frac{1}{2}[\sin (\frac{\pi }{7}-\frac{6\pi }{7})+\sin (\frac{\pi }{7}+\frac{6\pi }{7})]+\frac{1}{2}\sin \frac{\pi }{7}$

multiply by 2:

$\Rightarrow -\sin \frac{\pi }{7}+\sin \frac{3\pi }{7}-\sin \frac{3\pi }{7}+\sin \frac{5\pi }{7}-\sin \frac{5\pi }{7}+\sin \frac{7\pi }{7}+\sin \frac{\pi }{7}$

$-\sin \frac{\pi }{7}+\sin \pi +\sin \frac{\pi }{7}=0$ $\frac{3}{2}-1=\frac{1}{2}$

$\therefore \cos \frac{2\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{6\pi }{7}+\frac{1}{2}=0$

$\Rightarrow \cos \frac{2\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{6\pi }{7}+\frac{1}{2}+1=0+1=1$.