If $\cosh x = \operatorname{cosec} \theta$,then $\coth^2 \frac{x}{2} = $

If $4 \cos x + 3 \sin x = 5$,then find the value of $\tan x$.

If $\sin x + \sin y = 3(\cos y - \cos x),$ then the value of $\frac{\sin 3x}{\sin 3y}$ is

Let $S = \{\theta \in [0, 2\pi] : 8^{2 \sin^2 \theta} + 8^{2 \cos^2 \theta} = 16\}$. Then $n(S) + \sum_{\theta \in S} \left(\sec \left(\frac{\pi}{4} + 2\theta\right) \operatorname{cosec} \left(\frac{\pi}{4} + 2\theta\right)\right)$ is equal to.

The number of all possible values of $\theta$,where $0 < \theta < \pi$,for which the system of equations
$(y+z) \cos 3\theta = (xyz) \sin 3\theta$
$x \sin 3\theta = \frac{2 \cos 3\theta}{y} + \frac{2 \sin 3\theta}{z}$
$(xyz) \sin 3\theta = (y+2z) \cos 3\theta + y \sin 3\theta$
have a solution $(x_0, y_0, z_0)$ with $y_0 z_0 \neq 0$,is

If $u = \log \tan \left(\frac{\pi}{4} + \frac{\theta}{2}\right)$,then $\cosh u =$