If $\sec 4\theta – \sec 2\theta = 2$, then the general value of $\theta $ is

Let $\theta, 0 < \theta < \pi / 2$, be an angle such that the equation $x ^2+4 x \cos \theta+\cot \theta=0$ has equal roots for $x$. Then $\theta$ in radians is

If $S = \left\{ {x \in \left[ {0,2\pi } \right]:\left| {\begin{array}{*{20}{c}}
0&{\cos {\mkern 1mu} x}&{ – \sin {\mkern 1mu} x}\\
{\sin {\mkern 1mu} x}&0&{\cos {\mkern 1mu} x}\\
{\cos {\mkern 1mu} x}&{\sin {\mkern 1mu} x}&0
\end{array}} \right| = 0} \right\},$ then $\sum\limits_{x \in S} {\tan \left( {\frac{\pi }{3} + x} \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=(x y z) \sin 3 \theta $

$ x \sin 3 \theta=\frac{2 \cos 3 \theta}{y}+\frac{2 \sin 3 \theta}{z} $

$ (x y z) \sin 3 \theta=(y+2 z) \cos 3 \theta+y \sin 3 \theta$ have a solution $\left(\mathrm{x}_0, \mathrm{y}_0, \mathrm{z}_0\right)$ with $\mathrm{y}_0 \mathrm{z}_0 \neq 0$, is

$\cot \theta = \sin 2\theta (\theta \ne n\pi $, $n$ is integer), if $\theta = $