If $\tan A + \cot A = 2$,then the value of $\tan^{4} A + \cot^{4} A$ is:

If the sides of a right-angled triangle are ${\cos2\alpha + \cos2\beta + 2\cos(\alpha + \beta )}$ and ${\sin2\alpha + \sin2\beta + 2\sin(\alpha + \beta )}$,then the length of the hypotenuse is:

If $\pi < \alpha < \frac{3\pi}{2}$,then $\sqrt{\frac{1 - \cos \alpha}{1 + \cos \alpha}} + \sqrt{\frac{1 + \cos \alpha}{1 - \cos \alpha}} = $

All the pairs $(x, y)$ that satisfy the inequality $2^{\sqrt{\sin^2 x - 2 \sin x + 5}} \cdot \frac{1}{4^{\sin^2 y}} \leq 1$ also satisfy the equation

If $\alpha, \beta, \gamma, \delta$ are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity $k$,then the value of $4\sin \frac{\alpha}{2} + 3\sin \frac{\beta}{2} + 2\sin \frac{\gamma}{2} + \sin \frac{\delta}{2}$ is equal to

If $\theta$ is in the third quadrant,then $\sqrt{4 \sin ^4 \theta+\sin ^2 2 \theta}+4 \cos ^2\left(\frac{\pi}{4}-\frac{\theta}{2}\right)=$