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

In a triangle $ABC$,if $A = \frac{\pi}{4}$ and $\tan B \tan C = K$,then $K$ must satisfy:

$\frac{1+\sin \theta-\cos \theta}{1+\sin \theta+\cos \theta}$ 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 $x$).

$\tan \left( \frac{\pi }{4} + \theta \right) - \tan \left( \frac{\pi }{4} - \theta \right) = $

If $a \cos \theta + b \sin \theta = m$ and $a \sin \theta - b \cos \theta = n,$ then ${a^2} + {b^2} = $