If $(\sec A + \tan A)\,(\sec B + \tan B)\,(\sec C + \tan C)$ $ = \,(\sec A - \tan A)\,(\sec B - \tan B)\,(\sec C - \tan C),$ then each side is equal to

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

$\frac{{\cos A}}{{1 - \sin A}} = $

Prove that $\cot 4 x(\sin 5 x+\sin 3 x)=\cot x(\sin 5 x-\sin 3 x)$

If $\cos \theta = \frac{3}{5}$ and $\cos \phi = \frac{4}{5},$ where $\theta $ and $\phi $ are positive acute angles, then $\cos \frac{{\theta - \phi }}{2} = $

If $\alpha + \beta + \gamma = 2\pi ,$ then