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.

Match these equations to the graphs below and explain your reasoning.

\[y=\arcsin(\sin x)\]

\[y=\sin(\arcsin x)\]

\[y=\arccos(\cos x)\]

\[y=\cos(\arccos x)\]

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 \(\arccos(\cos x)=x\)?