The number of solutions of the equation $\sin 2x - 2 \cos x + 4 \sin x = 4$ in the interval $[0, 5\pi]$ 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:

If $\sin^2 \theta = \frac{1}{4},$ then the most general value of $\theta$ is

The number of solutions of the equation $\sin^{65}x - \cos^{65}x = -1$ for $x \in (-\pi, \pi)$ is:

Let $S = \{\theta \in (-2\pi, 2\pi) : \cos\theta + 1 = \sqrt{3} \sin\theta\}$. Then $\sum_{\theta \in S} \theta$ is equal to:

If $\cot \frac{x}{2} - \operatorname{cosec} \frac{x}{2} = \cot x$,then the values of $x$ are