Order doesn’t matter?

Geometry Level 3

True or false?

$\tan(\arctan x) = \arctan(\tan x)$

for all real numbers $x$ where $\tan x$ is defined.

True False

1 solution

Steven Yuan
Nov 30, 2017

The range of tangent is $(-\infty, \infty),$ but the range of arctan is $(-\frac{\pi}{2}, \frac{\pi}{2}).$ Therefore, the statement is false as one function can take on more values than the other. Below are the graphs of $\tan(\arctan x)$ and $\arctan(\tan x),$ which are colored red and blue respectively:

$Red: $\tan(\arctan x)$, Blue: $\arctan(\tan x)$$ Red: $\tan(\arctan x)$ , Blue: $\arctan(\tan x)$

Order doesn’t matter?

1 solution

0 pending reports