reserve th, th1, th2, th3 for Real;

theorem
  (cos(th1)+cos(th2))/(cos(th1)-cos(th2)) = cot((th1+th2)/2)*cot((th2- th1)/2)
proof
  (cos(th1)+cos(th2))/(cos(th1)-cos(th2)) = 2*(cos((th1+th2)/2)*cos((th1-
  th2)/2))/(cos(th1)-cos(th2)) by Th17
    .= 2*(cos((th1+th2)/2)*cos((th1-th2)/2)) /(-2*(sin((th1+th2)/2)*sin((th1
  -th2)/2))) by Th18
    .= 2*(cos((th1+th2)/2)*cos((th1-th2)/2)) /(2*((sin((th1+th2)/2)*(-sin((
  th1-th2)/2)))))
    .= 2*(cos((th1+th2)/2)*cos((th1-th2)/2)) /(2*((sin((th1+th2)/2)*sin(-((
  th1-th2)/2))))) by SIN_COS:31
    .= (2/2)*(cos((th1+th2)/2)*cos((th1-th2)/2) /(sin((th1+th2)/2)*sin(((th2
  -th1))/2))) by XCMPLX_1:76
    .= (cos((th1+th2)/2)/(sin((th1+th2)/2)) *(cos(-((th2-th1))/2)/sin((th2-
  th1)/2))) by XCMPLX_1:76
    .= cot((th1+th2)/2)*cot((th2-th1)/2) by SIN_COS:31;
  hence thesis;
end;
