reserve th, th1, th2, th3 for Real;

theorem
  cos(th1+th2)-cos(th1-th2) = -2*(sin(th1)*sin(th2))
proof
  cos(th1+th2)-cos(th1-th2) = (cos(th1)*cos(th2)-sin(th1)*sin(th2))-cos(
  th1-th2) by SIN_COS:75
    .= (cos(th1)*cos(th2)+-sin(th1)*sin(th2)) -(cos(th1)*cos(th2)+sin(th1)*
  sin(th2)) by SIN_COS:83
    .= -2*(sin(th1)*sin(th2));
  hence thesis;
end;
