$\cos ^2(x)+\cos ^2\left(x+\frac{\pi}{3}\right)+\cos ^2\left(x-\frac{\pi}{3}\right) = $

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

The number of solutions of the equation $32^{\tan^{2} x} + 32^{\sec^{2} x} = 81$ for $0 \leq x \leq \frac{\pi}{4}$ is:

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

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

If $\cos \alpha + \cos \beta = \frac{3}{2}$ and $\sin \alpha + \sin \beta = \frac{1}{2}$ and $\theta$ is the arithmetic mean of $\alpha$ and $\beta$,then $\sin 2\theta + \cos 2\theta$ is equal to