 Solution

Substitute $x={\sin }^{2}v$, with $0<v<\pi /2$, when integrating the target density. Since $dx=2\sin v\cos vdv$, its cumulative distribution function isInverting gives the arcsine distribution samplerThe inverse transform sampling proof in the root solution shows that it has the required probability density function. Use successive independent uniforms for successive samples. A draw at the endpoint one can be discarded if values strictly inside the support are desired; this event has probability zero under the ideal continuous uniform distribution.