Evaluate the expression: $\sum \frac{1}{1 + x^{a-b} + x^{a-c}}$

If $\cot \theta + \tan \theta = 3$ and $1 - \cos^2 \theta - \alpha \cos \theta = 0$,then

If $\cosh 2x = 199$,then $\coth x$ equals

If $\sinh x = \frac{12}{5}$,then $\sinh 3x + \cosh 3x = $

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

The smallest positive value (in degrees) of $\theta$ for which $\tan(\theta+100^{\circ})=\tan(\theta+50^{\circ}) \tan(\theta) \tan(\theta-50^{\circ})$ is valid,is (in $^{\circ}$)