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

Let $[x]$ denote the largest integer $\leq x$. If the number of solutions of $\sin x \sqrt{4 \cos ^2 x} = \frac{2+x-[x]}{1-x+[x]}$ is $k$,then for $x \in \left[\frac{\pi}{4}, \frac{\pi}{3}\right]$,the value of $k^{\tan^2 x}$

$\cos ^4 \frac{\pi}{12} + \cos ^4 \frac{5 \pi}{12} + \cos ^4 \frac{7 \pi}{12} + \cos ^4 \frac{11 \pi}{12} = $

If $\sin \theta + \sin 2\theta + \sin 3\theta = \sin \alpha$ and $\cos \theta + \cos 2\theta + \cos 3\theta = \cos \alpha$,then $\theta$ is equal to

If $\sin (y+z-x), \sin (z+x-y)$ and $\sin (x+y-z)$ are in $A$.$P$.,then

The expression $\cos^2 \alpha + \cos^2(\alpha + 120^\circ) + \cos^2(\alpha - 120^\circ)$ is equal to