If $\sec \theta - \tan \theta = \frac{1}{\sqrt{3}}$,the value of $\sec \theta \cdot \tan \theta$ is

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 $x$).

Let ${f_k}(x) = \frac{1}{k}(\sin^k x + \cos^k x)$ for $k = 1, 2, 3, ...$. Then for all $x \in R$,the value of $f_4(x) - f_6(x)$ is equal to

The value of $\log \tan 1^{\circ} + \log \tan 2^{\circ} + \log \tan 3^{\circ} + \ldots + \log \tan 89^{\circ}$ is equal to

The value of $\sin \frac{\pi }{14} \sin \frac{3\pi }{14} \sin \frac{5\pi }{14} \sin \frac{7\pi }{14} \sin \frac{9\pi }{14} \sin \frac{11\pi }{14} \sin \frac{13\pi }{14}$ is equal to

If $x \sin 45^{\circ} = y \operatorname{cosec} 30^{\circ}$,then $\frac{x^{4}}{y^{4}}$ is equal to (in $^{3}$)