I've come across different uses and meanings for $\sin^{-1}(x)$ by various authors:
- $\sin^{-1}(x) = \arcsin(x)$
- $\sin^{-1}(x) = \frac{1}{\sin(x)} = \csc(x)$
The Wikipedia article "Sin-1" says:
$\sin^{−1}y = \sin^{−1}(y)$, sometimes interpreted as $\arcsin(y)$ or arcsine of y, the compositional inverse of the trigonometric function sine (see below for ambiguity)
$\sin^{-1}x = \sin^{−1}(x)$, sometimes interpreted as $(sin(x))^{−1} = \frac{1}{\sin(x)} = \csc(x)$ or cosecant of x, the multiplicative inverse (or reciprocal) of the trigonometric function sine (see above for ambiguity)
When I look on Khan Academy, it tells me that $\sin^{-1}(x)$ does not stand for $\csc(x)$, instead:
If a number or variable is raised to the $-1$ power, then this refers to the multiplicative inverse, or the reciprocal. For example, $3^{-1} = \frac{1}{3}$. In general, if $a$ is a nonzero real number, then $a^{-1} = \frac{1}{a}$.
However, this is not the case for $\sin^{-1}(x)$. This is because the sine is a function, not a quantity!
In general, whenever you see a raised $-1$ after a function name, it refers to the inverse function.
Is it therefore considered a notational mistake to use a $-1$ exponent to denote the multiplicative inverse of a function?
Yes, I think it's fair to say it's wrong to write $\sin^{-1}(x)=1/\sin(x)$, unless you specify at the start that you're using the notation this way.
But the reasons you cite from Wikipedia and Khan are bogus! The reason the above is wrong is simply because it's a standard convention that $\sin^{-1}(x)=\arcsin(x)$. If the reasons given were valid they would show just as well that $\sin^2(x)=\sin(\sin(x))$, while in fact the standard convention there is $\sin^2(x)=(\sin(x))^2$.