What is $\arcsin(a+b)$ in terms of $\arcsin(a)$ and $\arcsin(b)$?

Question

How would I write $\sin^{-1}(a+b)$ in terms of $\sin^{-1}(a)$ and $\sin^{-1}(b)$? Assume that $\sin^{-1}(a+b)$ exists. Is it possible because I know you can easily write out $\sin^{-1}(a) + \sin^{-1}(b)$

Answer 1

Since $\arcsin x$ is odd what we can say is that

$$\arcsin a + \arcsin (-a)= \arcsin (a-a)=\arcsin 0 =0 $$

but there are not simple expression for the general $a$ and $b$.

One reason for this is that $\arcsin x$ is defined for $x\in[-1,1]$ thus if we consider, for example, $a=1$ and $b=\frac12$ we can define $\arcsin a$ and $\arcsin b$ but $\arcsin(a+b)$ is meaningless.