reserve th, th1, th2, th3 for Real;

theorem Th29:
  sin(th1)*sin(th2) = -(1/2)*(cos(th1+th2)-cos(th1-th2))
proof
  sin(th1)*sin(th2) = (cos(th1)*cos(th2)+sin(th1)*sin(th2) -cos(th1)*cos(
  th2)+sin(th1)*sin(th2))/2
    .= (cos(th1)*cos(-th2)+sin(th1)*sin(th2) -cos(th1)*cos(th2)+sin(th1)*sin
  (th2))/2 by SIN_COS:31
    .= (cos(th1)*cos(-th2)-sin(th1)*(-sin(th2)) -cos(th1)*cos(th2)+sin(th1)*
  sin(th2))/2
    .= (cos(th1)*cos(-th2)-sin(th1)*sin(-th2) -cos(th1)*cos(th2)+sin(th1)*
  sin(th2))/2 by SIN_COS:31
    .= (cos(th1+(-th2)) - cos(th1)*cos(th2)+sin(th1)*sin(th2))/2 by SIN_COS:75
    .= (cos(th1-th2)-(cos(th1)*cos(th2)-sin(th1)*sin(th2)))/2
    .= (cos(th1-th2)-cos(th1+th2))/(2/1) by SIN_COS:75
    .= -(1/2)*(cos(th1+th2)-cos(th1-th2));
  hence thesis;
end;
