reserve th, th1, th2, th3 for Real;

theorem Th32:
  cos(th1)*cos(th2)= (1/2)*(cos(th1+th2)+cos(th1-th2))
proof
  cos(th1)*cos(th2) = (1/2)*((cos(th1)*cos(th2)-sin(th1)*sin(th2)) +(cos(
  th1)*cos(th2)-(-sin(th1)*sin(th2))))
    .= (1/2)*(cos(th1+th2)+(cos(th1)*cos(th2)-sin(th1)*(-sin(th2)))) by
SIN_COS:75
    .= (1/2)*(cos(th1+th2)+(cos(th1)*cos(th2)-sin(th1)*sin(-th2))) by
SIN_COS:31
    .= (1/2)*(cos(th1+th2)+(cos(th1)*cos(-th2)-sin(th1)*sin(-th2))) by
SIN_COS:31
    .= (1/2)*(cos(th1+th2)+(cos(th1+(-th2)))) by SIN_COS:75
    .= (1/2)*(cos(th1+th2)+cos(th1-th2));
  hence thesis;
end;
