reserve th, th1, th2, th3 for Real;

theorem
  cos((th1+th2)/2)<>0 implies (sin(th1)-sin(th2))/(cos(th1)+cos(th2)) =
  tan((th1-th2)/2)
proof
  assume
A1: cos((th1+th2)/2)<>0;
  (sin(th1)-sin(th2))/(cos(th1)+cos(th2)) = 2*(cos((th1+th2)/2)*sin((th1-
  th2)/2))/(cos(th1)+cos(th2)) by Th16
    .= 2*(cos((th1+th2)/2)*sin((th1-th2)/2)) /(2*(cos((th1+th2)/2)*cos((th1-
  th2)/2))) by Th17
    .= (2/2)*((cos((th1+th2)/2)*sin((th1-th2)/2)) /(cos((th1+th2)/2)*cos((
  th1-th2)/2))) by XCMPLX_1:76
    .= (cos((th1+th2)/2)/cos((th1+th2)/2))*(sin((th1-th2)/2)/cos((th1-th2)/2
  )) by XCMPLX_1:76
    .= tan((th1-th2)/2) by A1,XCMPLX_1:88;
  hence thesis;
end;
