I have to prove that the Cauchy distribution is symmetric at 0. However, I'm not entirely sure how to do this. I'm given the problem: Suppose that a particle is fired from the origin in the $(x,y)$ plane in a straight line in a direction at random angle $\Phi$ to the $x$ axis and let $Y$ be the $y$-coordinate of the place where the particle hits the line $x=1$. Show that if $\Phi$ has uniform $(-\frac\pi2 , \frac\pi2)$ distribution, then $f_Y(y)=\frac{1}{\pi (1+y^2)}$ Show that the Cauchy distribution is symmetric about $0$. Firstly I would like to know a general strategy at "showing" things like this. I have seen a variety of problems that ask to show or derive something. Secondly how would I show that a probability distribution is symmetric about 0? A great description on how to do these would be much much appreciated. I can show what I have gotten so far:
I started by identifying that $f_\Phi(\phi)$ equals $\frac1\pi$ if $-\frac\pi2 \lt \phi \lt \frac\pi2$ and $0$ otherwise. Then by change of variables I divided the PDF of $\Phi$ divided by the absolute value of the derivative of $Y$. I think $y=arctan(x)$, therefore making the derivative $\frac1{(1+x^2)}$ but then once I put it all together I only came up with $\frac{(1+x^2)}{\pi}$. So I feel I'm close but I'm obviously doing something wrong. So if you took the time to read all this it is greatly appreciated.