The bivariate distribution formed by two independent
normalized Gaussians is rotationally symmetric (think about the
usual argument for evaluating the probability integral). The
quotient of two random variables $X$ and $Y$ is the tangent of the angle
between $(0,0)$ and $(X,Y)$ with the $x$-axis. If one has a rotationally
symmetric distribution for $X$ and $Y$ (with no point mass at the origin)
then $Y/X$ is a tangent of a uniformly distributed angle. This is the Cauchy
distribution.