Number of solutions of equation $secx = 1 + cosx + cos^2x + ........ \infty$ in $x \in [-50 \pi, 50 \pi]$ is -

All possible values of $\theta \in[0,2 \pi]$ for which $\sin 2 \theta+\tan 2 \theta>0$ lie in

The general solution of $\frac{{\tan \,2x\, - \,\tan \,x}}{{1\, + \,\tan \,x\,\tan \,2x}}\, = \,1$ is 

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 

If$\cos 6\theta + \cos 4\theta + \cos 2\theta + 1 = 0$, where $0 < \theta < {180^o}$, then $\theta  =$

For $x \in(0, \pi)$, the equation $\sin x+2 \sin 2 x-\sin 3 x=3$ has