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

could be used to denote the inverse cosine function, while probably instead refers to the reciprocal cosine function, i.e. the secant function.

In general, , but it's not unheard of for them to mean the same thing, especially in a context where something like is used explicitly to denote the inverse function. I'd be careful about assuming they are always the same.