# How to simplify $\arcsin(2\sin (x))$?

I know that $$\arcsin(\sin(x))$$ would be $$x$$ (in the right domain). But, I'm unsure how to handle the scalar multiplication. Any thoughts on how to proceed?

• There is no "nicer" expression for this. Commented Feb 6, 2021 at 8:26
• Why do you need such "simplification"? What are you trying to integrate? Commented Feb 6, 2021 at 8:27
• what an answer should give you? Commented Feb 6, 2021 at 8:34
• You could write it as $-i\ln\left(e^{ix}-e^{-ix}+\sqrt{e^{2ix}-1+e^{-2ix}}\right)$, but to me that looks worse. In fact, it also hints at why we can't do better than $\arcsin(2\sin x)$.
– J.G.
Commented Feb 6, 2021 at 13:09

You can write $$\arcsin(a\sin (x))=\sum_{n=0}^\infty \frac{ (2 n)!\,\, a^{2 n+1} }{4^n\,(2 n+1)\, (n!)^2}\,\sin ^{2 n+1}(x)$$
• @Sebastiano. At least I try (hard !). I felt in love with infinite series $64+$ years ago. In fact, take the infinite series of $\arcsin(t)$ and make $t=a\sin(x)$. Simple, no ? Cheers :-) Commented Feb 6, 2021 at 10:00