Question

Is there a difference between y=cos^-1(x) and y=(cos x)^-1?

Answer 1

$\cos^{-1}x$ could be used to denote the inverse cosine function, while $(\cos x)^{-1}$ probably instead refers to the reciprocal cosine function, i.e. the secant function.

In general, $\cos^{-1}x\neq(\cos x)^{-1}$, but it's not unheard of for them to mean the same thing, especially in a context where something like $\arccos x$ is used explicitly to denote the inverse function. I'd be careful about assuming they are always the same.