If $\frac{3\pi}{4} < \alpha < \pi,$ then $\sqrt{\csc^2 \alpha + 2\cot \alpha}$ is equal to

Given that $\cos \left( \frac{\alpha - \beta}{2} \right) = 2\cos \left( \frac{\alpha + \beta}{2} \right)$,then $\tan \frac{\alpha}{2} \tan \frac{\beta}{2}$ is equal to

If $\cos x + \cos^2 x = 1$,then the value of $\sin^2 x + \sin^4 x$ is

If $x = y \cos \frac{2\pi}{3} = z \cos \frac{4\pi}{3}$,then $xy + yz + zx = $

If $12a + 5b = 9$,where $a, b \in R$,then the minimum value of $a^2 + b^2$ is -

If $f(x) = \cos^2 x + \sec^2 x$,then