How to prove that $\tan x$ is strictly increasing without derivatives

Question

How can I show that the function $f(x) = \tan x$ is strictly increasing on $[-\pi/2, \pi/2]$ without using derivatives?

Answer 1

First understand that the tangent function is only strictly increasing in between its poles. So, within a region between adjacent poles, consider the difference formula (admittedly, from which the derivative may be derived):

$$\tan{x} - \tan{y} = \frac{\sin{x} \cos{y} - \cos{x} \sin{y}}{\cos{x} \cos{y}} = \frac{\sin{(x-y)}}{\cos{x} \cos{y}} $$

Note that, in between adjacent poles of the tangent function, the denominator is positive. Thus, when $x>y$, $\tan{x}>\tan{y}$.

Answer 2

Go back to the basics.

Recall that in the unit circle trigonometry, $\tan \theta$ is represented by the slope (gradient) of the line which makes an angle of $\theta$ with respect to the x-axis.

This is clearly an increasing function on the domain $[ - \frac {\pi}{2}, \frac {\pi}{2} ]$.