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

$\tan 9^\circ - \tan 27^\circ - \tan 63^\circ + \tan 81^\circ = $

If $\tan \theta + \cot \theta = 2$,then $\sin \theta$ is equal to

Factorize the expression: $\sec ^2 x+5 \tan x+5$

Let $a, b, c$ be three non-zero real numbers such that the equation $\sqrt{3} a \cos x + 2 b \sin x = c$,$x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ has two distinct real roots $\alpha$ and $\beta$ with $\alpha + \beta = \frac{\pi}{3}$. Then the value of $\frac{b}{a}$ 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: