The sum of the solutions in $(0, 2\pi)$ for the equation $\cos x \cos \left(\frac{\pi}{3}-x\right) \cos \left(\frac{\pi}{3}+x\right) = \frac{1}{4}$ is

The general solution of $\cot \frac{x}{2} - \cot x = \operatorname{cosec} \frac{x}{2}$ is

If $\sin^2 x + \sin x \cos x - 6\cos^2 x = 0$ and $-\frac{\pi}{2} < x < 0$,then the value of $\cos 2x$ is

If ${\left( {\frac{{\sin \theta }}{{\sin \phi }}} \right)^2} = \frac{{\tan \theta }}{{\tan \phi }} = 3,$ then the values of $\theta$ and $\phi$ are:

The number of solutions of the equation $\text{sgn}(\sin x) = \sin^2 x + 2\sin x + \text{sgn}(\sin^2 x)$ in the interval $\left[ -\frac{5\pi}{2}, \frac{7\pi}{2} \right]$ is (where $\text{sgn}(\cdot)$ denotes the signum function):

If the sum of all the solutions of the equation $8 \cos x \cdot \left( \cos \left( \frac{\pi}{6} + x \right) \cdot \cos \left( \frac{\pi}{6} - x \right) - \frac{1}{2} \right) = 1$ in the interval $[0, \pi]$ is $k\pi$,then $k$ is equal to: