If $\theta$ is an acute angle and $\tan^{2} \theta + \frac{1}{\tan^{2} \theta} = 2$,then the value of $\theta$ is (in $^{\circ}$)

If $\sin A, \cos A$ and $\tan A$ are in $G.P.$,then $\cos^3 A + \cos^2 A$ is equal to

If $\cos (\alpha + \beta ) = \frac{4}{5}, \sin (\alpha - \beta ) = \frac{5}{13}$ and $\alpha, \beta$ lie between $0$ and $\frac{\pi}{4}$,then $\tan 2\alpha = $

The value of the expression $1 - \frac{\sin^2 y}{1 + \cos y} + \frac{1 + \cos y}{\sin y} - \frac{\sin y}{1 - \cos y}$ is equal to

Let $f(x) = Ax^3 - Bx - \tan x \cdot \text{sgn}(x)$ be an even function for all $x \in \mathbb{R} - \left\{ (2n + 1) \frac{\pi}{2}, n \in \mathbb{Z} \right\}$, where $A = \sin^2 \alpha - \sin \alpha + \frac{1}{4}$ and $B = \tan^2 \alpha + \frac{2}{\sqrt{3}} \tan \alpha + \frac{1}{3}$. Then the number of values of $\alpha$ in $\left[ -\frac{3\pi}{2}, 2\pi \right]$ is (where $\text{sgn}(x)$ denotes the signum function of $x$).

$\sin 4\theta$ can be written as