If $2 \tan^2 \theta - 4 \sec \theta + 3 = 0$,then $2 \sec \theta =$

Let $[x]$ denote the largest integer $\leq x$. If the number of solutions of $\sin x \sqrt{4 \cos ^2 x} = \frac{2+x-[x]}{1-x+[x]}$ is $k$,then for $x \in \left[\frac{\pi}{4}, \frac{\pi}{3}\right]$,the value of $k^{\tan^2 x}$

If $2 \cos \theta + \sin \theta = 1$ $\left( \theta \neq \frac{\pi}{2} \right)$,then $7 \cos \theta + 6 \sin \theta$ is equal to

The value of $\tan 20^{\circ} \tan 80^{\circ} \cot 50^{\circ} = $

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 $\alpha, \beta, \gamma, \delta$ are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity $k$,then the value of $4\sin \frac{\alpha}{2} + 3\sin \frac{\beta}{2} + 2\sin \frac{\gamma}{2} + \sin \frac{\delta}{2}$ is equal to