Maclaurin series for $\arcsin(x)$

Question

I want to determine the Maclaurin series for $\arcsin(x)$. I know that $\arcsin'(x) = \frac{1}{\cos(\arcsin x))} = \frac{1}{\sqrt{1-x^2}}$. I also know that we can derive the polynomial by making use of the binomial theorem and integration. My question is if there is another, maybe more straightforward, way (e.g. by only making use of the definition of $\arcsin(x)$ (and its derivative) and the Taylor polynomial) to obtain the desired result.