If  $a = \sin \frac{\pi }{{18}}\sin \frac{{5\pi }}{{18}}\sin \frac{{7\pi }}{{18}}$ and $x$ is the solution of the equatioin $y = 2\left[ x \right] + 2$ and $y = 3\left[ {x - 2} \right] ,$ where $\left[ x \right]$ denotes the integral part of $x,$ then $a$ is equal to :-

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 solution of the equation $\sec \theta - {\rm{cosec}}\theta = \frac{4}{3}$ is

For which value of $x$  ;  $cosx > sinx,$ where $x\, \in \,\,\left( {\frac{\pi }{2}\,,\,\frac{{3\pi }}{2}} \right)$

If $\cot \theta + \tan \theta = 2{\rm{cosec}}\theta $, the general value of $\theta $ is

If $2{\sin ^2}\theta = 3\cos \theta ,$ where $0 \le \theta \le 2\pi $, then $\theta = $