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.

Try this (up to domains)...
Let $\theta = \arcsin\sqrt{x}$ and $\varphi = \arcsin(2x-1)$.  Therefore $\sin \theta = \sqrt{x}$ so $\sin^2\theta = x$.  Also $\sin \varphi = 2x-1 =2\sin^2\theta-1=-\cos(2\theta) = \sin(2\theta-\pi/2)$.  Therefore $\varphi = 2\theta - \pi/2$ so $\varphi+\pi/2 = 2\theta$ so
$$
\frac{\varphi}{\pi}+\frac{1}{2} = \frac{2}{\pi}\theta,
$$
which was to be proved.
