The number of solutions of the equation $x +2 \tan x =\frac{\pi}{2}$ in the interval $[0,2 \pi]$ is :

If equation in variable $\theta, 3 tan(\theta -\alpha) = tan(\theta + \alpha)$, (where $\alpha$ is constant) has no real solution, then $\alpha$ can be (wherever $tan(\theta - \alpha)$ & $tan(\theta + \alpha)$ both are defined)

$sin^{2n}x + cos^{2n}x$ lies between

The number of distinct solutions of the equation $\frac{5}{4} \cos ^2 2 x+\cos ^4 x+\sin ^4 x+\cos ^6 x+\sin ^6 x=2$ in the interval $[0,2 \pi]$ is

If $\cos ec\,\theta  = \frac{{p + q}}{{p - q}}$ $\left( {p \ne q \ne 0} \right)$, then $\left| {\cot \left( {\frac{\pi }{4} + \frac{\theta }{2}} \right)} \right|$ is equal to

The most general value of $\theta $ satisfying the equations $\tan \theta = - 1$ and $\cos \theta = \frac{1}{{\sqrt 2 }}$ is