The equation, $sin^2 \theta - \frac{4}{{{{\sin }^3}\,\,\theta \,\, - \,\,1}} = 1$$ -\frac{4}{{{{\sin }^3}\,\,\theta \,\, - \,\,1}}$ has :

Let $S=\left\{\theta \in(0,2 \pi): 7 \cos ^{2} \theta-3 \sin ^{2} \theta-2\right.$ $\left.\cos ^{2} 2 \theta=2\right\}$. Then, the sum of roots of all the equations $x ^{2}-2\left(\tan ^{2} \theta+\cot ^{2} \theta\right) x +6 \sin ^{2} \theta=0$ $\theta \in S$, is$...$

The solution of the equation ${\cos ^2}x - 2\cos x = $ $4\sin x - \sin 2x,$ $\,(0 \le x \le \pi )$ is

Let $P = \left\{ {\theta :\sin \,\theta  - \cos \,\theta  = \sqrt 2 \,\cos \,\theta } \right\}$ and $Q = \left\{ {\theta :\sin \,\theta  + \cos \,\theta  = \sqrt {2\,} \sin \,\theta } \right\}$ be two sets. Then

Find the general solution of $\cos ec\, x=-2$

If the equation $tan^4x -2sec^2x + [a]^2 = 0$ has atleast one solution, then the complete range of $'a'$ (where $a \in R$ ) is 
(Note : $[k]$ denotes greatest integer less than or equal to $k$ )