Inverse functions: what is the difference between $\tan^{-1}(x)$ and $\tan(x)^{-1}$?

Question

I’ve never really been taught about inverse functions, and I figured this is a pretty simple question, but I couldn’t find any explanation in my math textbook about this.

What is the difference between $\tan^{-1}$ and $\tan(x)^{-1}$?

Answer 1

$\tan^{-1}$ denotes the inverse tangent function, AKA the arc tangent (the angle the tangent of which is the given number). When applied to an argument, you spell

$$\tan^{-1}(x)=\arctan(x).$$

As far as I know, $$\tan(x)^{-1}$$ can be interpreted as the reciprocal of the tangent, i.e. the cotangent

$$\frac1{\tan(x)},$$ and it is safer to write $$(\tan(x))^{-1}=\cot(x).$$

Of course, this differs from

$$\tan(x^{-1})=\tan\left(\frac1x\right).$$

Answer 2

$\tan^{-1}$ would normally denote the FUNCTION $\mathbb{R}\longrightarrow (-\frac{\pi}{2}, \frac{\pi}{2})$ given by sending $x$ to the unique $y$ satisfying $\tan(y)=x$, while $\tan(x)^{-1}$ presumably stands for the real NUMBER, which is the multiplicative inverse $\frac{1}{\tan x}$ and is well-defined whenever $\tan(x)\neq 0$, i.e. whenever $x$ is not an integer multiple of $\pi$.