Transformation of Cauchy random variable If $X$ has a half-Cauchy distribution on (0,1) and we write $Y = 1/(1+ X^2)$, how can we show that $Y$ has a Be(0.5,0.5) (beta distribution)?
No idea how to start with this problem. I tried to integrate the density of the half Cauchy to get the cdf, but that equals 1.

Change of measures theorem: If $X$ is a random variable on some interval $I$ and $g:I \to J$ ($J$ is another interval) is either strictly increasing or strictly decreasing and differentiable with continuity with inverse $h,$ then the random variable $Y = g(X)$ on the interval $J$ has density (assuming $X$ has density $f_X$) $f_Y(y) = f_X(h(y))|h'(y)|.$
