For $x \in(0, \pi)$, the equation $\sin x+2 \sin 2 x-\sin 3 x=3$ has

Let $S=\left\{x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right): 9^{1-\tan ^2 x}+9^{\tan ^2 x}=10\right\}$ and $\beta=\sum_{x \in S} \tan ^2\left(\frac{x}{3}\right)$, then $\frac{1}{6}(\beta-14)^2$ is equal to

If sum of all the solutions of the equation $8\cos x \cdot \left( {\cos \left( {\frac{\pi }{6} + x} \right) \cdot \cos \left( {\frac{\pi }{6} – x} \right) – \frac{1}{2}} \right) = 1$ in $\left[ {0,\pi } \right]$ is $k\pi $then $k$ is equal to :

The general value of $\theta $ in the equation $2\sqrt 3 \cos \theta = \tan \theta $, is

If $\cos \,x = \frac{{2\cos y – 1}}{{2 – \cos y}},x,\,y\, \in \,\left( {0,\pi } \right),$ then $tan(x/2)cot(y/2) =$