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Okay, so you have inverse functions: $ x^{2^{\frac{1}{2}}} = x$

But you also have negative functions $ x^{2^{-2}} \neq x$

But when I look for inverse functions, they are usually defined like this:

$$ f^{-1} $$

However when I write $sin^{-1}(x)$, I am usually talking about $csc(x)$, which is not the inverse, which would be $asin(x)$.

Is there some better terminology for negative functions? What is now the actual inverse function?

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    *Reciprocals* . – 2017-01-31

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There's some finicky stuff that goes on with terminology.

We have $\sin^2(x)=(\sin(x))^2$, but $\sin^{-1}(x)$ is just inverse sin (or asin, arcsin, etc.)

If you want to say $\frac{1}{\sin x}$, the standard is to just use $csc(x).$

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    Or just use $\frac 1{\sin x}$, which is immediately understandable. – 2017-01-31
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    The standard in USA… – 2017-01-31
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    But isn't this simply wrong, because $sin^{1}(sin^{-1}(x)) \neq sin^{1 \cdot (-1)} (x)$ – 2017-01-31
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    Blame the convention. Yes, your argument makes sense, but for some reason people didn't think of inverse trig functions when they were inventing the $\sin^2(x)=(\sin x)^2$ gimmick. You can't really do anything about it :\ – 2017-01-31