If the random variable $X$ is standard Cauchy then so is $1/X$
Problem Prove that $X \in C(0,1) \Rightarrow 1/X \in C(0,1)$ where $C$ is the cauchy distribution.
Attempt 
I try to prove they have the same density function.
Question Is my proof correct?
$$f_{1/X}(x)=\left|-\frac1{x^2}\right|\cdot f_X\left(\frac1x\right)=\frac1{x^2}\cdot \frac1\pi\cdot \frac1{1+\left(\frac1x\right)^2}=\frac1\pi\cdot \frac1{1+x^2}=f_X(x)$$