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 $\tan \theta + \sec \theta = e^x$,then $\cos \theta$ equals

If $0 < x < \pi$ and $\cos x + \sin x = \frac{1}{2}$,then $\tan x$ is:

Consider the following two statements.
Statement $p$: The value of $\sin 120^\circ$ can be derived by taking $\theta = 240^\circ$ in the equation $2\sin \frac{\theta}{2} = \sqrt{1 + \sin \theta} - \sqrt{1 - \sin \theta}$.
Statement $q$: The angles $A, B, C$ and $D$ of any quadrilateral $ABCD$ satisfy the equation $\cos \left( \frac{1}{2}(A + C) \right) + \cos \left( \frac{1}{2}(B + D) \right) = 0$.
Then the truth values of $p$ and $q$ are respectively:

If $0 \le x \le \pi$ and $81^{\sin^2 x} + 81^{\cos^2 x} = 30$,then $x =$

The value of the series $\cos 12^{\circ} + \cos 84^{\circ} + \cos 132^{\circ} + \cos 156^{\circ}$ is