If $\alpha$ lies in the second quadrant,then $\sqrt{\frac{1-\sin \alpha}{1+\sin \alpha}}-\sqrt{\frac{1+\sin \alpha}{1-\sin \alpha}}=$

If $\cos A = \cos B \cos C$ and $A + B + C = \pi,$ then the value of $\cot B \cot C$ is

If $\sin \theta = - \frac{1}{\sqrt{2}}$ and $\tan \theta = 1$,then $\theta$ lies in which quadrant?

Let $f(x) = Ax^3 - Bx - \tan x \cdot \text{sgn}(x)$ be an even function for all $x \in \mathbb{R} - \left\{ (2n + 1) \frac{\pi}{2}, n \in \mathbb{Z} \right\}$, where $A = \sin^2 \alpha - \sin \alpha + \frac{1}{4}$ and $B = \tan^2 \alpha + \frac{2}{\sqrt{3}} \tan \alpha + \frac{1}{3}$. Then the number of values of $\alpha$ in $\left[ -\frac{3\pi}{2}, 2\pi \right]$ is (where $\text{sgn}(x)$ denotes the signum function of $x$).

If $\frac{3\pi}{4} < \alpha < \pi,$ then $\sqrt{\csc^2 \alpha + 2\cot \alpha}$ is equal to

Minimum value of $8 \cos^2 x + 18 \sec^2 x$ for all $x \in R$ wherever it is defined,is: