Hi, I am trying to show that when X and Y are two independent standard normal random variables, then the distribution of X/Y is the same as X/|Y|. (absolute value of Y)

I know that X/Y follows a Cauchy-distribution, using U = X + Y and V = X/Y and doing the transformation and using substitution method at the end for integral. But how do I prove that X/|Y| is also the same Cauchy distribution?