reserve th, th1, th2, th3 for Real;

theorem
  sin(th1+th2)+sin(th1-th2) = 2*(sin(th1)*cos(th2))
proof
  sin(th1+th2)+sin(th1-th2) = (sin(th1)*cos(th2)+cos(th1)*sin(th2))+sin(
  th1-th2) by SIN_COS:75
    .= (sin(th1)*cos(th2)+cos(th1)*sin(th2)) +(sin(th1)*cos(th2)---cos(th1)*
  sin(th2)) by SIN_COS:82
    .= 2*(sin(th1)*cos(th2));
  hence thesis;
end;
