Consider two independent random variables Xsub1 and Xsub2 having same Cauchy distribution
f(x)=1/(pi(1+x^2)) from negative infinity to positive infinity.
By using Theorem 7.2 to determine the joint probability density of Ysub1=Xsub1+Xsub2
and Ysub2=Xsub1-Xsub2 and then finding the marginal density of Ysub1.