${\sin ^2}\frac{\pi }{8} + {\sin ^2}\frac{{3\pi }}{8} + {\sin ^2}\frac{{5\pi }}{8} + {\sin ^2}\frac{{7\pi }}{8} = $

$\cos 20^\circ \cos 40^\circ \cos 80^\circ = $

If $A + B + C = \frac{{3\pi }}{2},$ then $\cos 2A + \cos 2B + \cos 2C = $

$\cos A + \cos (240^\circ + A) + \cos (240^\circ - A) = $

Prove that: $\cos 4 x=1-8 \sin ^{2} x \cos ^{2} x$

Let $S=\left\{x \in(-\pi, \pi): x \neq 0, \pm \frac{\pi}{2}\right\}$. The sum of all distinct solutions of the equation $\sqrt{3} \sec x+\operatorname{cosec} x+2(\tan x-\cot x)=0$ in the set $S$ is equal to