$\sin^2$ notation and uses of the alternative. So I was taking my calculus class and I was shocked by the following:
Apparently its a convention for 
$\sin^2(\alpha)=(\sin(\alpha))^2$
As opposed to what I thought made more sense which was
$\sin^2(\alpha)=\sin^2(\alpha)=\sin(\sin(\alpha)))$.
This made more sense to me because for two reasons.
reason 1 : $\sin^{-1}(\alpha)\neq(\sin(\alpha))^{-1}$ 
in other words it denotes the inverse function of sine, not the multiplicative inverse of sine of a number.
reason 2: 
writing $(\sin(\alpha))^n$ is a lot faster than writing $\underbrace{\sin(\sin(\dots(\sin(}_{n\text{ of these}}\alpha\underbrace{))\dots))}_{n\text{ of these}}$
So I guess that the justification for this is that the composition of the sine function is not very useful.
My question is, are there any interesting things that actually do use composition of trigonometric functions? is there an agreed upon definition for the composition of trigonometric functions?
Regards.

$\sin^2 x=(\sin x)^2=(\sin x)(\sin x)$ and that is not equal to $\sin(\sin x))$
Similarly: $(\sin x)^{-1}=\frac{1}{\sin x }$ but $\sin^{-1}x=\arcsin x$ Lastly, $\sin(x^2)=\sin(x*x)$.
