How do I go about proving this series expansion $$\frac{\sin^{-1}(x)}{\sqrt{1-x^{2}}}=\sum_{r=0}^{\infty}\frac{x^{2r+1}4^{r}}{(2r+1)\binom{2r}{r}}?$$
I have used this expansion while solving other problems but I tried and could not prove this expansion. I know the individual expansions of $\sin^{-1} (x)$ and $\frac{1}{\sqrt{1-x^{2}}}$.
...and they both contain the $\binom{2r}{r}$ in the numerator. I know that this must have something to do with the differentiation of the series expansion of $(\sin^{-1} (x))^{2}$. But in that case I do not know how to derive the series for $(\sin^{-1} (x))^{2}$. So if someone can provide a step by step proof of either one of the series then I would be really grateful. If the proof for one of the series is provided then I can prove the other series by differentiation or integration(obviouslt I have to show the conditions for term by term integration or differentiation are satisfied....but I will worry about that later).
