reserve th, th1, th2, th3 for Real;

theorem
  cos(th1)<>0 & sin(th2)<>0 implies tan(th1)-cot(th2)= -cos(th1+th2)/(
  cos(th1)*sin(th2))
proof
  assume cos(th1)<>0 & sin(th2)<>0;
  then
  tan(th1)-cot(th2)= (sin(th1)*sin(th2)---cos(th1)*cos(th2))/(cos(th1)*sin
  (th2)) by XCMPLX_1:130
    .= (-(cos(th1)*cos(th2)-sin(th1)*sin(th2)))/(cos(th1)*sin(th2))
    .= -(cos(th1)*cos(th2)-sin(th1)*sin(th2))/(cos(th1)*sin(th2)) by
XCMPLX_1:187
    .= -cos(th1+th2)/(cos(th1)*sin(th2)) by SIN_COS:75;
  hence thesis;
end;
