Inverse trigonometric expansion related question

Question

I know expansions for $\sin^{-1}(x)+\sin^{-1}(y)$, but does there exists any expansion for $\sin^{-1}(x \pm y)$ if not then what is the reason?

Answer 1

Every reasonable function of two variables has a Taylor expansion, which is a sum of homogeneous polynomials of increasing degree.

Your first function makes it trivial as it is the sum of two one-variable functions, and the polynomials are of the form $c_k(x^{2k+1}+y^{2k+1})$.

The second function isn't much more complex, as you can write it as the sum of $c_k(x\pm y)^{2k+1}$. These can be developed using the binomial formula

$$c_k(x\pm y)^{2k+1}=c_k\sum_{j=0}^{2k+1}\binom nj x^{j}(\pm y)^{2k+1-j}.$$