The number of solutions of $\tan x + \sec x = 2 \cos x$ in $[0, 2 \pi]$ is

If $\sin \left(\frac{\pi}{4} \cot \theta\right) = \cos \left(\frac{\pi}{4} \tan \theta\right)$,then the general solution of $\theta$ is

If $\cos(\alpha - \beta) = 1$ and $\cos(\alpha + \beta) = \frac{1}{e}$,where $-\pi < \alpha, \beta < \pi$,then the total number of ordered pairs $(\alpha, \beta)$ is:

The number of solutions of $\cos 2 \theta = \sin \theta$ in $(0, 2 \pi)$ is

The common principal solution of the equations $\sin \theta = -\frac{1}{2}$ and $\tan \theta = \frac{1}{\sqrt{3}}$ is

Let $S = \left[-\pi, \frac{\pi}{2}\right) - \left\{-\frac{\pi}{2}, -\frac{\pi}{4}, -\frac{3\pi}{4}, \frac{\pi}{4}\right\}$. Then the number of elements in the set $\{\theta \in S : \tan \theta(1 + \sqrt{5} \tan(2\theta)) = \sqrt{5} - \tan(2\theta)\}$ is: