The number of solutions of the equation $\sin x=$ $\cos ^{2} x$ in the interval $(0,10)$ is

Let $S=\{x \in R: \cos (x)+\cos (\sqrt{2} x)<2\}$, then

The number of solutions of the pair of equations $ 2 \sin ^2 \theta-\cos 2 \theta=0 $, $ 2 \cos ^2 \theta-3 \sin \theta=0$ in the interval $[0,2 \pi]$ is

If $0 \le x < 2\pi $ , then the number of real values of $x,$ which satisfy the equation  $\cos x + \cos 2x + \cos 3x + \cos 4x = 0$ is  . .  .

The number of  $x \in  [0,2\pi ]$  for which $\left| {\sqrt {2\,{{\sin }^4}\,x\, + \,18\,{{\cos }^2}\,x}  - \,\sqrt {2\,{{\cos }^4}\,x\, + \,18\,{{\sin }^2}\,x} } \right| = 1$ is

If $12{\cot ^2}\theta - 31\,{\rm{cosec }}\theta + {\rm{32}} = {\rm{0}}$, then the value of $\sin \theta $ is