Standard normal density function
Solution 1:
Hint: For every real number $u$ in $(0,1)$, the random variable $Xu+Y\sqrt{1-u^2}$ is standard normal. Hence, for every $(0,1)$ valued random variable $U$ independent of $(X,Y)$, the random variable $XU+Y\sqrt{1-U^2}$ is standard normal. Use this for $U=1/\sqrt{1+Z^2}$.