reserve th, th1, th2, th3 for Real;

theorem Th12:
  cos(th1)<>0 & cos(th2)<>0 & cos(th3)<>0 implies cos(th1+th2+th3)
= cos(th1)*cos(th2)*cos(th3) *(1-tan(th2)*tan(th3)-tan(th3)*tan(th1)-tan(th1)*
  tan(th2))
proof
  assume that
A1: cos(th1)<>0 and
A2: cos(th2)<>0 and
A3: cos(th3)<>0;
  cos(th1+th2+th3)=cos(th1+(th2+th3))
    .=cos(th1)*cos(th2+th3)-sin(th1)*sin(th2+th3) by SIN_COS:75
    .=cos(th1)*(cos(th2)*cos(th3)-sin(th2)*sin(th3)) -sin(th1)*sin(th2+th3)
  by SIN_COS:75
    .=cos(th1)*(cos(th2)*cos(th3))-cos(th1)*(sin(th2)*sin(th3)) -(sin(th1))*
  (sin(th2)*cos(th3)+cos(th2)*sin(th3)) by SIN_COS:75
    .=cos(th1)*(cos(th2)*cos(th3))-cos(th1)*((cos(th2)*tan(th2))*sin(th3)) -
  (sin(th1)*(sin(th2)*cos(th3)+cos(th2)*sin(th3))) by A2,Th6
    .=cos(th1)*(cos(th2)*cos(th3)) -cos(th1)*((cos(th2)*tan(th2))*(cos(th3)*
  tan(th3))) -(sin(th1)*(sin(th2)*cos(th3)+cos(th2)*sin(th3))) by A3,Th6
    .=cos(th1)*(cos(th2)*cos(th3)) -cos(th1)*((cos(th2)*tan(th2))*(cos(th3)*
tan(th3))) -((cos(th1)*tan(th1))*(sin(th2)*cos(th3) + cos(th2)*sin(th3))) by A1
,Th6
    .=cos(th1)*(cos(th2)*cos(th3)) -cos(th1)*((cos(th2)*tan(th2))*(cos(th3)*
tan(th3))) -((cos(th1)*tan(th1))*((cos(th2)*tan(th2))*cos(th3) + cos(th2)*sin(
  th3))) by A2,Th6
    .=(cos(th1)*cos(th2))*cos(th3) -((cos(th1)*cos(th2))*cos(th3))*(tan(th2)
*tan(th3)) -((cos(th1)*tan(th1))*(((cos(th2)*cos(th3))*tan(th2)) +cos(th2)*(cos
  (th3)*tan(th3)))) by A3,Th6
    .=cos(th1)*cos(th2)*cos(th3) *(1-tan(th2)*tan(th3)-tan(th3)*tan(th1)-tan
  (th1)*tan(th2));
  hence thesis;
end;
