Where do the inverses stand?

Geometry Level 2

Which of the given signs should fill the box if $x$ is any real number?
$\large \arctan\left(\tan x\right) \boxed{\phantom{0}} \tan\left(\arctan x\right)$

Consider the domain of $x$ in this case as $\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$

Cannot be determined > = <

1 solution

Ashish Menon
Jun 5, 2016

For $x \in \left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$ ,
$\large \arctan\left(\tan x\right) \color{#3D99F6}{\boxed{=}} \tan\left(\arctan x\right) = x$

This is incorrect.
$\arctan(\tan(x))$ is not always equal to x.
The range of $\arctan(x)$ is $(\dfrac{-\pi}{2}, \dfrac{\pi}{2} )$ .
So ,
$\arctan(\tan(x)) = x$ only if $x \in(\dfrac{-\pi}{2}, \dfrac{\pi}{2} )$
For $x \in(\dfrac{\pi}{2}, \dfrac{3\pi}{2} )$
$\arctan(\tan(x)) = x - \pi$

A Former Brilliant Member - 5 years ago

Thanks, I will specify that in my question

Ashish Menon - 5 years ago

Where do the inverses stand?

1 solution

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