$\sin (\pi + \theta )\sin (\pi - \theta )\csc^2 \theta = $

The value of ${\tan ^2}\theta + {\cot ^2}\theta$ is:

If $\tan \theta = \frac{\sin \alpha - \cos \alpha}{\sin \alpha + \cos \alpha}$,then $\sin \alpha + \cos \alpha$ and $\sin \alpha - \cos \alpha$ must be equal to

If $7 \sin^{2} \theta + 3 \cos^{2} \theta = 4$,then the value of $\tan \theta$ is (where $\theta$ is acute).

If $\Delta ABC$ is right-angled at $B$,$AB = 6$ units,and $\angle C = 30^{\circ}$,then $AC$ is equal to (in units):

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: