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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?

ReinstateMonica3167040
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    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 '20 at 00:27
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    You need to use context clues to see which is meant. I prefer writing $\arcsin$ for this reason. – Jair Taylor Jan 28 '20 at 00:29
  • Does this mess apply to all functions? – ReinstateMonica3167040 Jan 28 '20 at 00:31
  • I found a similar discussion at https://math.stackexchange.com/questions/1117986/why-is-arcsin-represented-with-the-1-notation but I couldn't find a clear answer how to know what would be intended by something combining both such as sin⁻²(x) – ReinstateMonica3167040 Jan 28 '20 at 00:34
  • 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 '20 at 00:37

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Apparently it's just like natural languages, and this is just an annoying exception. While I don't know what the edge case of a negative power that is not 1, I understand that the two notations don't play nicely, and aren't related.

ReinstateMonica3167040
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