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I want to find a law for distributing, so to speak, the arcsin function over the product of two or more variables, sort of like how $\log(XY) = \log(X)+\log(Y)$. I have tried to find solutions online but I can only find other trig identities.

Peter Phipps
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Darcy Sutton
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    The logarithm does not work like that. It's $\log xy = \log x + \log y.$ For your actual issue, start with $\alpha = \arcsin xy.$ Now use some inverse functions – Sean Roberson Dec 27 '22 at 19:41
  • @SeanRoberson oh sorry yeah that was a stupid mistake of mine. Ok I'll have a shot at it thanks. – Darcy Sutton Dec 27 '22 at 19:42
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    $\sin^{-1}(xy)$ is a natural logarithm. – Тyma Gaidash Dec 27 '22 at 19:49
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    You can maybe have $$\arcsin X\pm\arcsin Y=\arcsin(X\sqrt{1-Y^2}\pm Y\sqrt{1-X^2})$$ or something like that, possibly with some additional restrictions on $X,Y$ (on top of $-1\le X,Y\le 1$). In fact, see: https://math.stackexchange.com/questions/672575/proof-for-the-formula-of-sum-of-arcsine-functions-arcsin-x-arcsin-y I have, however, never seen a formula with $\arcsin(XY)$ broken up in a useful way. –  Dec 27 '22 at 20:05

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