On my calculator $\tan^{-1}$ is used to calculate the $\arctan$, but $\tan^{-1}$ actually is $\cot$. $\cot$ and $\arctan$ are not the same thing though. Am I missing something or is the labeling of my Casio fx-991ES really wrong?
To make the question more clear: Is $\arctan = \tan^{-1}$ correct?
Calculators have to save space on the labels, therefore $\tan^{-1}$ is more convenient than $\arctan$.
Moreover, the notation $f^{-1}$ conventionally denotes the functional inverse of $f$. It's rare to write $f^{-1}$ meaning $1/f$.
On my calculator, for example, $\sin^{-1}$ and $\cos^{-1}$ are used in place of the more correct (in my opinion, since less confusing) $\arcsin$ and $\arccos$.
The notation $\tan^{-1}$ is often reserved for the canonical functional inverse of the $\tan$ version rather than its reciprocal. You're right that it's confusing, but that's the conventional mathematical notation.
Indeed this is ambiguous, but the reason for that is that $f^{-1}$ is the inverse element of $f$ by composition law ($(f \circ f)(x) = f(f(x))$ : $$f \circ f^{-1} = \text{id}$$ Where id is the identity function, the neutral of $(F(\mathbb{R},\mathbb{R}), \circ)$.
