The general solution of the trigonometric equation $\tan \theta = \cot \alpha$ is

The general solution of $\cot \frac{x}{2} - \cot x = \operatorname{cosec} \frac{x}{2}$ is

If $\cos \theta + \cos 2\theta + \cos 3\theta = 0$,then the general value of $\theta$ is

If $\frac{1 - \cos 2\theta}{1 + \cos 2\theta} = 3$,then the general value of $\theta$ is

When $a$ is irrational,the number of solutions satisfying the equation $1+\sin^2(ax)=\cos(x)$ is

If the solution of the equation $\log _{\cos x} \cot x+4 \log _{\sin x} \tan x=1$,where $x \in \left(0, \frac{\pi}{2}\right)$,is $\sin ^{-1}\left(\frac{\alpha+\sqrt{\beta}}{2}\right)$,where $\alpha, \beta$ are integers,then $\alpha+\beta$ is equal to: