 Solution

Condition on the fair coin, taken independent of $U$. For the squared uniform distribution, the cumulative distribution function of $U^2$ is $\sqrt{x}$ on $0<x<1$, giving probability density function $1/(2\sqrt{x})$. Reflection about $1/2$ gives the probability density function of $1-U^2$ as $1/(2\sqrt{1-x})$. Averaging the two conditional probability density functions produces this mixture model:There are no atoms at the endpoints; each integrable singularity contributes half the total mass. Equivalently the cumulative distribution function is $[\sqrt{x}+1-\sqrt{1-x}]/2$ on the interval.