If $\tan \left(\frac{\pi}{4}+\alpha\right)=\tan ^3\left(\frac{\pi}{4}+\beta\right)$, then $\tan (\alpha+\beta) \cot (\alpha-\beta)=$

If $0 \leq x \leq 3$ and $0 \leq y \leq 3$, then the number of solutions $(x, y)$ of the equation $\left(\sqrt{\sin ^2 x-\sin x+\frac{1}{2}}\right) 2^{\sec ^2 y}=1$ is

Statement I In the interval $[0,2 \pi]$, the number of common solutions of the equations $2 \sin ^2 \theta-\cos 2 \theta=0$ and $2 \cos ^2 \theta-3 \sin \theta=0$ is two.

Statement II The number of solutions of $2 \cos ^2 \theta-3 \sin \theta=0$ in $[0, \pi]$ is two.

The number of solutions of the equation $4 \cos 2 \theta \cos 3 \theta=\sec \theta$ in the interval $[0,2 \pi]$ is