Infinite series expansion of $\arcsin (x)$ and $\arccos (x)$

Question

How to find the infinite series expansion of $\arcsin (x)$ and $\arccos (x)$?

Answer 1

HINT:

As $$\frac{d(\arcsin x)}{dx}=(1-x^2)^{-\dfrac12}$$

expand the right hand side & integrate either side

FInally set $x=0\implies \arcsin x=0 $ to find the arbitrary constant

We can use $\arcsin x+\arccos x=\dfrac\pi2$ for $\arccos x$