[tex]\cos^{-1}x[/tex] could be used to denote the inverse cosine function, while [tex](\cos x)^{-1}[/tex] probably instead refers to the reciprocal cosine function, i.e. the secant function.
In general, [tex]\cos^{-1}x\neq(\cos x)^{-1}[/tex], but it's not unheard of for them to mean the same thing, especially in a context where something like [tex]\arccos x[/tex] is used explicitly to denote the inverse function. I'd be careful about assuming they are always the same.
