reserve th, th1, th2, th3 for Real;

theorem
  sin(th1)<>0 & sin(th2)<>0 implies cot(th1)-cot(th2)= -sin(th1-th2)/(
  sin(th1)*sin(th2))
proof
  assume sin(th1)<>0 & sin(th2)<>0;
  then
  cot(th1)-cot(th2)= (cos(th1)*sin(th2)---cos(th2)*sin(th1))/(sin(th1)*sin
  (th2)) by XCMPLX_1:130
    .= (-(sin(th1)*cos(th2)-cos(th1)*sin(th2)))/(sin(th1)*sin(th2))
    .= (-(sin(th1-th2)))/(sin(th1)*sin(th2)) by SIN_COS:82
    .= -sin(th1-th2)/(sin(th1)*sin(th2)) by XCMPLX_1:187;
  hence thesis;
end;
