If $2 \sin \left(\frac{\pi}{2}\right) = x^{2} + \frac{1}{x^{2}}$,then the value of $\left(x - \frac{1}{x}\right)$ is

If $\sin A = \frac{1}{\sqrt{10}}$ and $\sin B = \frac{1}{\sqrt{5}}$,where $A$ and $B$ are positive and acute angles,then $A + B = $

Given the equation $4x^2 + 4(a - 1)x + (1 - 2a) = 0$ has roots $\sin \theta$ and $\cos \theta$ where $0 < \theta < \frac{\pi}{2}$,then the maximum value of $(a + \sin \theta)$ is:

If $\sec A = a + \left(\frac{1}{4a}\right)$,then $\sec A + \tan A =$

In a $\Delta ABC$,$\angle B = \frac{\pi}{3}$,$\angle C = \frac{\pi}{4}$ and $D$ divides $BC$ internally in the ratio $1:3$,then $\frac{\sin \angle BAD}{\sin \angle CAD}$ is equal to

The value of $\frac{\cos 2 \theta}{1-\sin 2 \theta}=$