The number of solutions of $|\cos x|=\sin x$, such that $-4 \pi \leq x \leq 4 \pi$ is.

If $\tan \theta + \tan 2\theta + \sqrt 3 \tan \theta \tan 2\theta = \sqrt 3 ,$ then

If $\cos \,\alpha  + \cos \,\beta  = \frac{3}{2}$ and $\sin \,\alpha  + \sin \,\beta  = \frac{1}{2}$ and $\theta $ is the the arithmetic mean of $\alpha $ and $\beta $ , then $\sin \,2\theta  + \cos \,2\theta $ is equal to 

The number of solutions that the equation $sin5\theta cos3\theta  = sin9\theta cos7\theta $ has in $\left[ {0,\frac{\pi }{4}} \right]$ is

The general value of $\theta $ satisfying the equation $\tan \theta + \tan \left( {\frac{\pi }{2} – \theta } \right) = 2$, is