If $\frac{3\pi}{4} < \alpha < \pi,$ then $\sqrt{\csc^2 \alpha + 2\cot \alpha}$ is equal to

The value of $e^{\log _{10} \tan 1^{\circ}+\log _{10} \tan 2^{\circ}+\log _{10} \tan 3^{\circ}+\ldots+\log _{10} \tan 89^{\circ}}$ is

Let $\theta, \phi \in [0, 2\pi]$ be such that $2 \cos \theta(1-\sin \phi) = \sin^2 \theta \left(\tan \frac{\theta}{2} + \cot \frac{\theta}{2}\right) \cos \phi - 1$,$\tan (2\pi - \theta) > 0$ and $-1 < \sin \theta < -\frac{\sqrt{3}}{2}$. Then $\phi$ cannot satisfy

If $\cos x = \tan y$,$\cot y = \tan z$ and $\cot z = \tan x$,then $\sin x$ equals to

$\sec {50^o} + \tan {50^o}$ is equal to

Let $\alpha, \beta$ and $\gamma$ be such that $0 < \alpha < \beta < \gamma < 2 \pi$. For any $x \in \mathbb{R}$,if $\cos (x+\alpha)+\cos (x+\beta)+\cos (x+\gamma)=0$,then $\tan (\gamma-\alpha) = $