If $tanA + cotA = 4$, then $tan^4A + cot^4A$ is equal to

The number of solutions of the equation $32^{\tan ^{2} x}+32^{\sec ^{2} x}=81,0 \leq x \leq \frac{\pi}{4}$ is :

If $S = \left\{ {x \in \left[ {0,2\pi } \right]:\left| {\begin{array}{*{20}{c}}
0&{\cos {\mkern 1mu} x}&{ - \sin {\mkern 1mu} x}\\
{\sin {\mkern 1mu} x}&0&{\cos {\mkern 1mu} x}\\
{\cos {\mkern 1mu} x}&{\sin {\mkern 1mu} x}&0
\end{array}} \right| = 0} \right\},$ then $\sum\limits_{x \in S} {\tan \left( {\frac{\pi }{3} + x} \right)} $ is equal to

The roots of the equation $1 - \cos \theta = \sin \theta .\sin \frac{\theta }{2}$ is

The number of pairs $(x, y)$ satisfying the equations $\sin x + \sin y = \sin (x + y)$ and $|x| + |y| = 1$ is

$\cot \theta = \sin 2\theta (\theta \ne n\pi $, $n$ is integer), if $\theta = $