What is the minimum value of $a \sec x + b \csc x$ for $0 < a < b$ and $0 < x < \pi/2$?

If $\alpha \in \left( 0, \frac{\pi}{2} \right)$,then $\sqrt{x^2 + x} + \frac{\tan^2 \alpha}{\sqrt{x^2 + x}}$ is always greater than or equal to

Let $x, y, z \in [0, 1]$. Then the maximum value of $\sqrt{|x-y|} + \sqrt{|y-z|} + \sqrt{|z-x|}$ is

$a, b, c$ are the sides of a scalene triangle $ABC$. If angles $\alpha, \beta, \gamma$ lie between $0$ and $\pi$ such that $\cos \alpha = \frac{a}{b+c}, \cos \beta = \frac{b}{c+a}$ and $\cos \gamma = \frac{c}{a+b}$,then $\tan^2 \frac{\alpha}{2} + \tan^2 \frac{\beta}{2} + \tan^2 \frac{\gamma}{2} =$

The minimum value of $5 \tan^2 \alpha + \frac{9}{\tan^2 \alpha} + 4 \sec^2 \alpha$ is:

Let $y=4 \sin^2 \theta - \cos 2 \theta$. If $l$ and $m$ are the minimum and maximum values of $y$ respectively,then