reserve th, th1, th2, th3 for Real;

theorem
  cos(th1)<>0 & cos(th2)<>0 implies tan(th1)-tan(th2)= sin(th1-th2)/(cos
  (th1)*cos(th2))
proof
  assume cos(th1)<>0 & cos(th2)<>0;
  then
  tan(th1)-tan(th2)= (sin(th1)*cos(th2)-sin(th2)*cos(th1))/(cos(th1)*cos (
  th2)) by XCMPLX_1:130
    .= sin(th1-th2)/(cos(th1)*cos(th2)) by SIN_COS:82;
  hence thesis;
end;
