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
$2 \sqrt{3}$
$\frac{3}{2}$
$\sqrt{3}$
$\frac{1}{2}$
If $tan(\pi sin \theta)$ $= cot(\pi cos \theta)$, then $\left| {\cot \left( {\theta - \frac{\pi }{4}} \right)} \right|$ is -
The number of distinct solutions of the equation $\log _{\frac{1}{2}}|\sin x|=2-\log _{\frac{1}{2}}|\cos x|$ in the interval $[0,2 \pi],$ is
If $\cos \theta + \sec \theta = \frac{5}{2}$, then the general value of $\theta $ is
If $\sin 3\alpha = 4\sin \alpha \sin (x + \alpha )\sin (x - \alpha ),$ then $x = $
$\tan \,{20^o}\cot \,{10^o}\cot \,{50^o}$ is equal to