Question

Show that $f\left(x\right)=\tan x$ is an inreasing function on $(-\pi /2,\pi /2)$

Answer 1

We have

$f\left(x\right)=\tan x\Rightarrow f^{\prime }\left(x\right)={\sec }^{2}x$

Now, $-\pi /2<x<\pi /2\Rightarrow \sec x>0\Rightarrow {\sec }^{2}x>0\Rightarrow f^{\prime }\left(x\right)>0$

So, $f\left(x\right)$ is increasing on $(-\pi /2,\pi /2)$