Why isn't $\arctan(\tan x)=x$?

Question

I am in high school and my teacher happened to mention that $ \arctan(\tan x) $ isn't ALWAYS $x$ but that $\tan(\arctan x)$ is always $x$. Why the difference between the two ?

Answer 1

$\arctan(\tan x)=x$ $\mbox{ }\ \ \ $if $-\dfrac{\pi}2<x<\dfrac{\pi}2$.

Otherwise its not.

Answer 2

Since $\tan x$ is periodic, any function of the form $f(\tan x)$ is also periodic, so isn't the identity function. Consider $\arctan\tan\frac{5\pi}{4}=\frac{\pi}{4}$.

Answer 3

That is because $\tan x$ isn't a bijective function. To obtain a bijective function one has to consider its restriction to some relevant interval on which it becomes a bijection – in practice the interval $(-\tfrac\pi2,\tfrac\pi 2)$. So by definition $$y=\arctan x\iff \tan y =x\quad\textbf{and}\quad -\tfrac\pi 2<y<\tfrac\pi 2$$

Thus, we have $$\arctan\bigl(\tan\tfrac{5\pi}4\bigr)=\arctan 1=\tfrac\pi4.$$

Answer 4

Here are the graphs of $\arctan(\tan(x))$(green) and $x$(blue). Notice that they only overlap in the region $\frac{-\pi}{2}$ and $\frac{\pi}{2}$