2tan(y-x)=2(tan(y)-tan(x))/(1+tan(y)tan(x)).
If tan(y)=2tan(x)+cot(x), then:
2tan(y-x)=2(2tan(x)-tan(x)+cot(x))/(1+(2tan(x)+cot(x))tan(x)),
2tan(y-x)=2(tan(x)+cot(x))/(1+2tan2(x)+1),
2tan(y-x)=2(tan(x)+cot(x))/(2(1+tan2(x)),
2tan(y-x)=(tan(x)+cot(x))/(1+tan2(x)),
2tan(y-x)=cot(x)(tan2(x)+1)/(1+tan2(x))=cot(x) QED