If $\alpha + \beta = \frac{\pi }{2}$ and $\beta + \gamma = \alpha ,$ then $\tan \,\alpha $ equals

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 $A + B + C = \pi ,$ then $\cos \,\,2A + \cos \,\,2B + \cos \,\,2C = $

If $cos A = {3\over 4} , $ then $32\sin \left( {\frac{A}{2}} \right)\sin \left( {\frac{{5A}}{2}} \right) = $

$cot 5^o$ -$tan5^o$ -$2$ $tan10^o$ -$4$ $tan 20^o$ -$8$ $cot40^o$ is equal to

If $\alpha$, $\beta$,$\gamma$ are positive number such that $\alpha + \beta = \pi$  and $\beta  + \gamma = \alpha$, then $tan\ \alpha$ is equal to - (where $\gamma  \ne n\pi ,n \in I$ )