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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?

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    $\begingroup$ 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 $\sin^{k}(x)$. $\endgroup$ – Jair Taylor Jan 28 at 0:27
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    $\begingroup$ You need to use context clues to see which is meant. I prefer writing $\arcsin$ for this reason. $\endgroup$ – Jair Taylor Jan 28 at 0:29
  • $\begingroup$ Does this mess apply to all functions? $\endgroup$ – User that is not a user Jan 28 at 0:31
  • $\begingroup$ 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) $\endgroup$ – User that is not a user Jan 28 at 0:34
  • $\begingroup$ 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... $\endgroup$ – Jair Taylor Jan 28 at 0:37
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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.

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