Functions of Cauchy distributions

Question

If X is a random variable which is distributed Cau(0,1) (i.e. standard cauchy) with density $$f(x)=\frac{1}{\pi}\frac{1}{1+x^2}$$

Then how would $\frac{2X}{1-X^2}$ be distributed if we know that $$\frac{2X}{1-X^2}=tan(2\alpha)$$

Thank you