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

The value of the expression $(sinx + cosecx)^2 + (cosx + secx)^2 - (tanx + cotx)^2$ wherever defined is equal to

$\cos ^{2} 5^{\circ}+\cos ^{2} 10^{\circ}+\cos ^{2} 15^{\circ}+\cdots+\cos ^{2} 90^{\circ}=$

The value of $\sin 780^{\circ} \sin 480^{\circ} + \cos 240^{\circ} \cos 300^{\circ} = $

If $A + B + C = 180^o,$ then the value of $(\cot B + \cot C)(\cot C + \cot A)(\cot A + \cot B)$ 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 $x$).