For $0 \leq P, Q \leq \frac{\pi}{2},$ if $\sin P + \cos Q = 2,$ then the value of $\tan \left(\frac{P + Q}{2}\right)$ is equal to

It is known that $\sin \beta = \frac{4}{5}$ and $0 < \beta < \pi$. Then the value of $\frac{\sqrt{3} \sin(\alpha + \beta) - \frac{2}{\cos(\pi/6)} \cos(\alpha + \beta)}{\sin \alpha}$ is:

$\frac{\cot A}{1-\tan A}+\frac{\tan A}{1-\cot A} = ?$

If $\frac{\sqrt{2} \sin \alpha}{\sqrt{1+\cos 2 \alpha}}=\frac{1}{7}$ and $\sqrt{\frac{1-\cos 2 \beta}{2}}=\frac{1}{\sqrt{10}}$ where $\alpha, \beta \in (0, \frac{\pi}{2})$,then $\tan (\alpha+2 \beta)$ is equal to

If $0 \leq x \leq \frac{\pi}{2}$, then the number of values of $x$ for which $\sin x - \sin 2x + \sin 3x = 0$ is

The geometric mean of $\tan 1^{\circ}, \tan 2^{\circ}, \ldots, \tan 89^{\circ}$ is