Is $f(f^{-1}(x))=x \quad \land \quad f^{-1}(f(x))=x$ true for all inverse trigonometric functions?

Question

Based on: $f(f^{-1}(x))=x \quad \quad\land\quad \quad f^{-1}(f(x))=x$

Are all the following definitions true?

• $\arcsin(\sin x)=x$

• $\sin(\arcsin x) = x$

• $\arccos (\cos x) = x$

• $\cos (\arccos x) = x$

• $\arctan(\tan x) = x$

• $\tan(\arctan x) = x$

• $\text{arccot} (\cot x) = x$

• $\cot (\text{arccot} x)=x$

That's make everything easier if they all were true. Thanks for clarification.

Answer 1

No.

To elaborate a bit on the previous answer, inverse trig functions have a limited range, so they’ll give the angle within their allowed quadrants. For example:

$$\arcsin\bigg(\sin \frac{7\pi}{6}\bigg) = -\frac{\pi}{6}$$

This happens since the range of $\arcsin x$ is $\big[-\frac{\pi}{2}, \frac{\pi}{2}\big]$.

Also, $\tan x$ and $\cot x$ are not defined for all real numbers, unlike their inverse counterparts. You might want to learn the domain and range of the $6$ main trig functions then switch their domains and ranges for their inverses. (Knowing these can help greatly.)

Answer 2

The ones like $\sin(\arcsin(x)=x$ [with the inverse function "inside"] are true, not the others. Try experimenting with large angles for $x.$