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)= (cos(th1)*cos(th2)+sin(th1)*sin(th2))/(cos(th1)*sin (
  th2)) by XCMPLX_1:116
    .= cos(th1-th2)/(cos(th1)*sin(th2)) by SIN_COS:83;
  hence thesis;
end;
