Notation for powers of trigonometric functions 
Possible Duplicate:
$\arcsin$ written as $\sin^{-1}(x)$ 

When I learnt the trig identity $\sin^2\theta + \cos^2\theta \equiv 1$, I learnt that $\sin^2\theta = (\sin\theta)^2$.
So why isn't $\sin^{-1}\theta = (\sin\theta)^{-1} = \dfrac{1}{\sin\theta}$?
Because $\csc\theta = \dfrac{1}{\sin\theta} $, but $\csc\theta \ne \sin^{-1}\theta$
How can these two same notations, just with different numbers, mean different things?
 A: It is a notational thing, there is nothing mathematically wrong here.
We denote the inverse of a function $f$ by $f^{-1}$. So when you see sin$^{-1}(x)$, this means "the inverse sin function", not the reciprocal.
But also sin$^2 (x)$ is just notation for $(\text{sin}(x))^2$, there is no mathematical reason why they are equal, it is by definition.
I know it is slightly confusing but it is notation that has stuck through time...
A: It's only a matter of convention. $f^{-1}$ is used to mean the inverse function of $f$, pretty much always. Only with $\cos$, $\sin$ and other trigonometric functions we usually multiply those together, or square or cube them. That's why it becomes useful to write $\cos^2 x$ as a shortcut for $(\cos x)^2$, for example. But it's just notation (that can be confusing at times).
A: I seem to recall reading that Gauss said that the notation $\sin^2\theta$ ought to mean $\sin(\sin(\theta))$, and similarly for other powers.  That would be consistent with using $\sin^{-1}$ to mean the inverse function.
So $\sin^{-1}$ means the inverse function because $\sin^3\theta$ ought to mean $\sin\sin\sin\theta$.
A: Because $\sin^{-1}$ is thought of more as the inverse function. So let's say $f = \sin$. Then the inverse function if $f^{-1} = \sin^{-1}$. Also, $\sin^{-1}$ can be also written as $\arcsin$.
A: This universally observed convention is just an example of overloading of notation, where the meaning of notation depends upon context.
