Kurs:Mathematik für Anwender (Osnabrück 2019-2020)/Teil I/Repetitorium/16/reciprocal and inverse of trigonometric functions/Studentenfrage/Antwort

The inverse map to a bijective map ${\displaystyle {}f\colon M\rightarrow N}$ is the map ${\displaystyle {}f^{-1}\colon N\rightarrow M}$ such that ${\displaystyle {}f^{-1}(y)}$ is the unique element ${\displaystyle {}x}$ for which ${\displaystyle {}f(x)=y}$.

The reciprocal of a function ${\displaystyle {}f\colon I\rightarrow \mathbb {R} }$ on the other hand is a function ${\displaystyle {}g\colon J\subseteq I\longrightarrow \mathbb {R} }$, ${\displaystyle {}x\longmapsto {\frac {1}{f(x)}}}$. This is of course only defined if ${\displaystyle {}f(x)\neq 0}$ for every ${\displaystyle {}x\in J}$. For example the reciprocal of the cosine is called the secant function and the reciprocal of the sine is called the cosecant function. I guess your confusion with the inverse functions comes because the values of the reciprocal functions are the inverses of the values of their counterparts. But the inverses of the values have nothing to do with the inverse of a function.

I think the names for these functions are not important, as you can just write ${\displaystyle {}{\frac {1}{\cos x}}}$ and ${\displaystyle {}{\frac {1}{\sin x}}}$ instead. The only reciprocal function we introduced is the cotangens ${\displaystyle {}\cot x}$ whose values are the inverses of the tangens. Maybe there is a merit to it, because it is zero where the tangens is not defined.
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