No. of solution of equation $sin^{65}x\, -\, cos^{65}x =\, -1$ is, if $x \in (-\pi , \pi )$

Common roots of the equations $2{\sin ^2}x + {\sin ^2}2x = 2$ and $\sin 2x + \cos 2x = \tan x,$ are

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

If the equation $2tan\ x \ sin\ x -2 tan\ x + cos\ x = 0$ has $k$ solutions in $[0,k \pi]$, then number of integral values of $k$ is-

Number of solutions of the equation $2^x + x = 2^{sin \ x} +  \sin x$ in $[0,10\pi ]$ is -

The number of real solutions of the equation $2 \sin 3 x+\sin 7 x-3=0$, which lie in the interval $[-2 \pi, 2 \pi]$ is