The number of all values of $\theta$ in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ satisfying the equation $(1-\tan \theta)(1+\tan \theta) \sec ^2 \theta+2 \tan ^2 \theta=0$ is

Let $S=\left\{x \in(-\pi, \pi) \mid x \neq 0, \pm \frac{\pi}{2}\right\}$. The sum of all distinct solutions of the equation $\sqrt{3} \sec x+\operatorname{cosec} x+2(\tan x-\cot x)=0$ in the set S is equal to

The number of roots of the equation, $(81)^{\sin ^2 x}+(81)^{\cos ^2 x}=30$ in the interval $[0, \pi]$, is equal to

Let $2 \sin ^2 x+3 \sin x-2>0$ and $x^2-x-2<0$. ( $x$ is measured in radians). The $x$ lies in the interval