The value of $\tan 100^{\circ} + \tan 125^{\circ} + \tan 100^{\circ} \tan 125^{\circ} =$ ?

If $A = \sin^2 x + \cos^4 x$,then for all real $x :$

If $\tan \alpha = \frac{1}{7}$ and $\sin \beta = \frac{1}{\sqrt{10}}$ where $0 < \alpha, \beta < \frac{\pi}{2}$,then $2\beta$ is equal to:

If $\tan \alpha = \frac{x^2 - x}{x^2 - x + 1}$ and $\tan \beta = \frac{1}{2x^2 - 2x + 1}$ $(x \ne 0, 1)$,where $0 < \alpha, \beta < \frac{\pi}{2}$,then $\tan(\alpha + \beta)$ has the value equal to:

If $5 \cos \theta + 12 \sin \theta = 13$ and $0^{\circ} < \theta < 90^{\circ}$,then the value of $\sin \theta$ is:

If $A$ lies in the second quadrant and $B$ lies in the third quadrant and $\cos A = -\frac{\sqrt{3}}{2}$ and $\sin B = -\frac{3}{5}$,then find the value of $\frac{2 \tan B + \sqrt{3} \tan A}{\cot^2 A + \cos B}$.