What is the value of the expression $\sin\frac{2\pi}{7}\sin\frac{4\pi}{7}+\sin\frac{4\pi}{7}\sin\frac{8\pi}{7}+\sin\frac{8\pi}{7}\sin\frac{2\pi}{7}$? This is rather a simple problem that I'm posting ; looking forward not for the solution of it but the different ways it could be solved.
What is the value of 
$\sin\frac{2\pi}{7}\sin\frac{4\pi}{7}+\sin\frac{4\pi}{7}\sin\frac{8\pi}{7}+\sin\frac{8\pi}{7}\sin\frac{2\pi}{7}$ 
Do I just use 
$\cos(a-b)-\cos(a+b) = 2\sin(a)\sin(b)$

$\sin\frac{2\pi}{7}\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}\sin\frac{8\pi}{7}$
$\sin \frac{8\pi}{7} = \sin \frac{-6 \pi}{7}$
-> $\sin\frac{2\pi}{7}\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}\sin\frac{-6\pi}{7}$

$\sin a \sin (b-c) + \sin b \sin (c-a)+ \sin c \sin (a-b)$ is always equal to "$0$".

Can you see it?
