If $\sqrt{3}\left(\cos ^{2} x\right)=(\sqrt{3}-1) \cos x+1,$ the number of solutions of the given equation when $x \in\left[0, \frac{\pi}{2}\right]$ is

If $\frac{{\tan 3\theta - 1}}{{\tan 3\theta + 1}} = \sqrt 3 $, then the general value of $\theta $ is

If $2(\sin x - \cos 2x) - \sin 2x(1 + 2\sin x)2\cos x = 0$ then

The general solution of $\sin x - \cos x = \sqrt 2 $, for any integer $n$ is

The number of solutions to the equation $\cos ^4 x+\frac{1}{\cos ^2 x}=\sin ^4 x+\frac{1}{\sin ^2 x}$ in the interval $[0,2 \pi]$ is

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