Values of $\theta$ $(0^\circ < \theta < 360^\circ)$ satisfying $\text{cosec } \theta + 2 = 0$ are

The number of values of $x$ satisfying $2\sin^2(2x) = 2\cos^2(8x) + \cos(10x)$ in the interval $x \in \left[ -\frac{\pi}{4}, \frac{\pi}{4} \right]$ is:

If $m$ and $n$ respectively are the numbers of positive and negative values of $\theta$ in the interval $[-\pi, \pi]$ that satisfy the equation $\cos 2 \theta \cos \frac{\theta}{2} = \cos 3 \theta \cos \frac{9 \theta}{2}$,then $mn$ is equal to $.............$.

If $x \in(-\pi, \pi)$ then the number of solutions of the equation $2 \sin x \sin 3 x \sin 5 x+\sin 5 x \cos 4 x=0$ 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 $2 \sin x + \cos x = 3$ is