What is difference between $\sin \theta ^2$ and $\sin ^2\theta$? What is the meaning of $\sin \theta\ \times$ $\sin \theta$?
$\endgroup$ 61 Answer
$\begingroup$Because there are many cases where we want powers of trig functions, we define $\sin^2(x)=(\sin(x))^2$. It doesn't have to be that way. For more general functions, we often define $f^2(x)=f(f(x))$, which is not the same thing. We also define $\sin^{-1}(x)$ as the inverse function of $\sin(x)$ (though many denote it $\arcsin(x)$), with the range restricted to make it a function. By analogy with $\sin^2(x)$ it should be $\frac 1{\sin(x)}$. The $\arcsin$ version is in accord with the usual notation of $f^{-1}(x)$ being the inverse function of $f(x)$. Yes, it is inconsistent, but it is useful, so you need to get used to it.
$\endgroup$ 2
