If $\sin A = n \sin (A + 2B)$,then $\tan (A + B) =$

The value of the expression $\sqrt{3} \operatorname{cosec} 20^{\circ}-\sec 20^{\circ}$ is equal to

Let $\alpha, \beta$ and $\gamma$ be such that $0 < \alpha < \beta < \gamma < 2 \pi$. For any $x \in \mathbb{R}$,if $\cos (x+\alpha)+\cos (x+\beta)+\cos (x+\gamma)=0$,then $\tan (\gamma-\alpha) = $

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

The number of all possible triplets $(a_1, a_2, a_3)$ such that $a_1 + a_2 \cos 2x + a_3 \sin^2 x = 0$ for all $x$ is

If $\sin 2\theta + \sin 2\phi = \frac{1}{2}$ and $\cos 2\theta + \cos 2\phi = \frac{3}{2}$,then $\cos^2(\theta - \phi) =$