Food for thought

## Problem

Take a look at the three graphs below.

One of the graphs shows $y=\arctan(\tan x).$ Another shows $y=\tan (\arctan x)$. Which graphs are these, and why? (The small open circles on the graphs indicate that these points are excluded.)

Please note that $\arctan$ and $\tan^{-1}$ mean the same thing and both are widely used in books and online. Similarly, $\arcsin$ and $\sin^{-1}$ mean the same as one another, as do $\arccos$ and $\cos^{-1}$.

If you want to know more about $\arctan x$, $\arcsin x$ and $\arccos x$ before working on this task, you may find it helpful to look at Inverse trigonometric functions.

For which values of $x$ is $\tan(\arctan x)=x$? What about $\arctan(\tan x)=x$?

What can you say about the solutions of similar equations, such as $\sin(\arcsin x)=x$ or $\arccos(\cos x)=x$?