How to prove this arcsine identity? In the top of this Wikipedia article for the arcsine distribution
it states that
$$\frac{2}{\pi} \arcsin(\sqrt{x}) = \frac{\arcsin(2x-1)}{\pi} + \frac{1}{2}$$
Why is this true? I haven't been able to derive this.

Here's one way:
$$
\frac{2}{\pi} \arcsin(\sqrt{x}) \overset{\text{??}}{=} \frac{\arcsin(2x-1)}{\pi} + \frac{1}{2}
$$
Differentiate both sides:
$$
\frac2\pi\cdot\frac{1/(2\sqrt{x})}{\sqrt{1-x}} \overset{\text{??}}{=} \frac1\pi \frac{2}{\sqrt{1-(2x-1)^2}}
$$
These are routinely seen to be equal, so the expressions above them must differ by a constant.  Plugging in $x=0$ then finishes it off.
One could also just use trigonometric identities.  Maybe I'll post more later....
