Given that $\cos \left( {\frac{{\alpha - \beta }}{2}} \right) = 2\cos \left( {\frac{{\alpha + B}}{2}} \right)$, then $\tan \frac{\alpha }{2}\tan \frac{\beta }{2} $ is equal to

For any $\theta \, \in \,\left( {\frac{\pi }{4},\frac{\pi }{2}} \right)$, the expression $3\,{\left( {\sin \,\theta  - \cos \,\theta } \right)^4} + 6{\left( {\sin \,\theta  + \cos \,\theta } \right)^2} + 4\,{\sin ^6}\,\theta $ equals

If $A + B + C = {180^o},$ then the value of $(\cot B + \cot C)$ $(\cot C + \cot A)\,\,(\cot A + \cot B)$ will be

If $A, B, C$ are angles of a triangle, then $\sin 2A + \sin 2B - \sin 2C$ is equal to

Let $\frac{\pi}{2} < x < \pi$ be such that $\cot x=\frac{-5}{\sqrt{11}}$. Then $\left(\sin \frac{11 x}{2}\right)(\sin 6 x-\cos 6 x)+\left(\cos \frac{11 x}{2}\right)(\sin 6 x+\cos 6 x)$ is equal to

If $\theta = 3\, \alpha$ and $sin\, \theta =$ $\frac{a}{{\sqrt {{a^2}\,\, + \,\,{b^2}} }}$. The value of the expression , $a \,cosec\, \alpha - b \,sec\, \alpha$ is