reserve th, th1, th2, th3 for Real;

theorem Th11:
  cos(th1)<>0 & cos(th2)<>0 & cos(th3)<>0 implies sin(th1+th2+th3)
= cos(th1)*cos(th2)*cos(th3) *(tan(th1)+tan(th2)+tan(th3)-tan(th1)*tan(th2)*tan
  (th3))
proof
  assume that
A1: cos(th1)<>0 and
A2: cos(th2)<>0 and
A3: cos(th3)<>0;
  sin(th1+th2+th3) = sin(th1+(th2+th3))
    .=sin(th1)*cos(th2+th3)+cos(th1)*sin(th2+th3) by SIN_COS:75
    .=sin(th1)*(cos(th2)*cos(th3)-sin(th2)*sin(th3)) + cos(th1)*sin(th2+th3)
  by SIN_COS:75
    .=sin(th1)*(cos(th2)*cos(th3))-sin(th1)*(sin(th2)*sin(th3)) +cos(th1)*(
  sin(th2)*cos(th3)+cos(th2)*sin(th3)) by SIN_COS:75
    .=(cos(th1)*tan(th1))*(cos(th2)*cos(th3))-sin(th1)*(sin(th2)*sin(th3)) +
  (cos(th1)*(sin(th2)*cos(th3))+cos(th1)*(cos(th2)*sin(th3))) by A1,Th6
    .=(cos(th1)*tan(th1))*(cos(th2)*cos(th3)) -(cos(th1)*tan(th1))*(sin(th2)
*sin(th3)) +(cos(th1)*(sin(th2)*cos(th3))+cos(th1)*(cos(th2)*sin(th3))) by A1
,Th6
    .=(cos(th1)*tan(th1))*(cos(th2)*cos(th3)) -(cos(th1)*tan(th1))*((cos(th2
)*tan(th2))*sin(th3)) +(cos(th1)*(sin(th2)*cos(th3))+cos(th1)*(cos(th2)*sin(th3
  ))) by A2,Th6
    .=(cos(th1)*tan(th1))*(cos(th2)*cos(th3)) -(cos(th1)*tan(th1))*((cos(th2
)*tan(th2))*(cos(th3)*tan(th3))) +(cos(th1)*(sin(th2)*cos(th3))+cos(th1)*(cos(
  th2)*sin(th3))) by A3,Th6
    .=(cos(th1)*tan(th1))*(cos(th2)*cos(th3)) -(cos(th1)*tan(th1))*((cos(th2
)*tan(th2))*(cos(th3)*tan(th3))) +(cos(th1)*((cos(th2)*tan(th2))*cos(th3))+cos(
  th1)*(cos(th2)*sin(th3))) by A2,Th6
    .=(cos(th1)*cos(th2))*(cos(th3)*tan(th1)) -(cos(th1)*cos(th2)*(tan(th1)*
tan(th2)))*(cos(th3)*tan(th3)) +(cos(th1)*((cos(th2)*tan(th2))*cos(th3)) +cos(
  th1)*(cos(th2)*(cos(th3)*tan(th3)))) by A3,Th6
    .=cos(th1)*cos(th2)*cos(th3) *(tan(th1)+tan(th2)+tan(th3)-tan(th1)*tan(
  th2)*tan(th3));
  hence thesis;
end;
