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 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:

The sum of all $x \in [0, \pi]$ which satisfy the equation $\sin x + \frac{1}{2} \cos x = \sin^2(x + \frac{\pi}{4})$ is

If $\sin 2\theta = \cos 3\theta$ and $\theta$ is an acute angle,then $\sin \theta$ is equal to

For $n \in Z$,the general solution of the equation $(\sqrt{3} - 1) \sin \theta + (\sqrt{3} + 1) \cos \theta = 2$ is

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