If $\theta = 3\alpha$ and $\sin \theta = \frac{a}{\sqrt{a^2 + b^2}}$,then the value of the expression $a \operatorname{cosec} \alpha - b \sec \alpha$ is

If $A = 580^\circ$,then which one of the following is true?

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 $x$).

If $\cos A = \cos B \cos C$ and $A + B + C = \pi,$ then the value of $\cot B \cot C$ is

If $\cos \theta = \frac{x^{2}-y^{2}}{x^{2}+y^{2}}$,then the value of $\cot \theta$ is equal to (where $0^{\circ} \leq \theta \leq 90^{\circ}$)

If $(\sec A + \tan A)(\sec B + \tan B)(\sec C + \tan C) = (\sec A - \tan A)(\sec B - \tan B)(\sec C - \tan C),$ then each side is equal to