The number of solutions of the equation $|\cot x|=\cot x+\frac{1}{\sin x}$ in the interval $[0,2 \pi]$ is

If $e ^{\left(\cos ^{2} x+\cos ^{4} x+\cos ^{6} x+\ldots \ldots \infty\right) \log _{e} 2}$ satisfies the equation $t ^{2}-9 t +8=0,$ then the value of $\frac{2 \sin x}{\sin x+\sqrt{3} \cos x}\left(0 < x < \frac{\pi}{2}\right)$ is

If $\sin \,\theta  + \sqrt 3 \cos \,\theta  = 6x - {x^2} - 11,x \in R$ , $0 \le \theta  \le 2\pi $ , then the equation has solution for

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

The sides of a triangle are $\sin \alpha ,\,\cos \alpha $ and $\sqrt {1 + \sin \alpha \cos \alpha } $ for some $0 < \alpha < \frac{\pi }{2}$. Then the greatest angle of the triangle is.....$^o$

If $\cos 2\theta + 3\cos \theta = 0$, then the general value of $\theta $ is