# Raising functions to powers

I have learned that sin²(x) = (sin x)². Why does it not generalize such that sin⁻¹(x) = (sin x)⁻¹?

P.S. I understand that they are not equal, sin⁻¹(x) is arcsin x, but (sin x)⁻¹ is csc x.

EDIT: What would sin⁻²(x) be?

• This is just an unfortunate choice of notation I'm afraid. There's not a logical reason, really, just two different, contradictory meanings for $k$ in the notation $\sin^{k}(x)$. – Jair Taylor Jan 28 at 0:27
• You need to use context clues to see which is meant. I prefer writing $\arcsin$ for this reason. – Jair Taylor Jan 28 at 0:29
• Does this mess apply to all functions? – User that is not a user Jan 28 at 0:31
• I found a similar discussion at math.stackexchange.com/questions/1117986/… but I couldn't find a clear answer how to know what would be intended by something combining both such as sin⁻²(x) – User that is not a user Jan 28 at 0:34
• It just depends on context. Sometimes authors use $f^k$ to mean the pointwise product, and sometimes they use it to mean the composition $f \circ f \circ \ldots \circ f$ ($k$ times), e.g. in a permutation group. But hopefully the two notations are not used on the same set in the same chapter/article... – Jair Taylor Jan 28 at 0:37