The expression,$\frac{{\tan \,\left( {{\textstyle{{3\,\pi } \over 2}}\,\, - \,\,\alpha } \right)\,\,\,\cos \,\left( {{\textstyle{{3\,\pi } \over 2}}\,\, - \,\,\alpha } \right)}}{{\cos \,(2\,\pi \,\, - \,\alpha )}}$ $+ cos \left( {\alpha \,\, - \,\,\frac{\pi }{2}} \right) \,sin (\pi -\alpha ) + cos (\pi +\alpha ) sin \,\left( {\alpha \,\, - \,\,\frac{\pi }{2}} \right)$ when simplified reduces to :

$\cos 2(\theta + \phi ) - 4\cos (\theta + \phi )\sin \theta \sin \phi + 2{\sin ^2}\phi = $

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

The value of $cos\, \frac{\pi }{{10}} \,cos\, \frac{2\pi }{{10}} \,cos\,\frac{4\pi }{{10}}\, cos\,\frac{8\pi }{{10}}\, cos\,\frac{16\pi }{{10}}$ is 

In triangle $ABC$, the value of $\sin 2A + \sin 2B + \sin 2C$ is equal to

Suppose $\theta $ and $\phi  (\ne 0)$ are such that $sec\,(\theta  + \phi ),$ $sec\,\theta $ and $sec\,(\theta  - \phi )$ are in $A.P.$ If $cos\,\theta  = k\,cos\,( \frac {\phi }{2})$ for some $k,$ then $k$ is equal to