For alpha I've used a, for beta I've used b, and for theta I've used ø.
sin(a-b)=sin(a)cos(b)-cos(a)sin(b). Let tan(a)=X and tan(b)=Y then sin(a)=X/√(1+X^2), cos(a)=1/√(1+X^2), sin(b)=Y/√(1+Y^2), cos(b)=1/√(1+Y^2) and tan(a)/tan(b)=X/Y. So x=X and y=Y.
sin(a-b)=x/√((1+x^2)(1+y^2))-y/√((1+x^2)(1+y^2))=(x-y)/√((1+x^2)(1+y^2)).
But:
sinø=sin(a+b)=x/√((1+x^2)(1+y^2))+y/√((1+x^2)(1+y^2))=
(x+y)/√((1+x^2)(1+y^2)).
Therefore sin(a-b)/sin(a+b)=sin(a-b)/sinø=(x-y)/(x+y);
sin(a-b)=((x-y)/(x+y))*sinø QED.