reserve th, th1, th2, th3 for Real;

theorem
  sin(th1) <> 0 & sin(th2) <> 0 & sin(th3) <> 0 implies cot(th1+th2+th3)
= (cot(th1)*cot(th2)*cot(th3)-cot(th1)-cot(th2)-cot(th3)) /(cot(th2)*cot(th3)+
  cot(th3)*cot(th1)+cot(th1)*cot(th2)-1)
proof
  assume that
A1: sin(th1) <> 0 and
A2: sin(th2) <> 0 and
A3: sin(th3) <> 0;
A4: sin(th1)*sin(th2) <> 0 by A1,A2;
  cot(th1+th2+th3) = (cos(th1+th2)*cos(th3)-sin(th1+th2)*sin(th3)) /sin(
  th1+th2+th3) by SIN_COS:75
    .= ((cos(th1)*cos(th2)-sin(th1)*sin(th2))*cos(th3)-sin(th1+th2)*sin(th3)
  ) /sin(th1+th2+th3) by SIN_COS:75
    .= (cos(th3)*(cos(th1)*cos(th2)-sin(th1)*sin(th2)) -(sin(th3)*(sin(th1)*
  cos(th2)+cos(th1)*sin(th2)))) /sin(th1+th2+th3) by SIN_COS:75
    .= ((cos(th3)*(cos(th1)*cos(th2)-sin(th1)*sin(th2)) -(sin(th3)*(sin(th1)
*cos(th2)+cos(th1)*sin(th2)))) /(sin(th1)*sin(th2)*sin(th3))) /(sin(th1+th2+th3
  )/(sin(th1)*sin(th2)*sin(th3))) by A3,A4,XCMPLX_1:55
    .= ((cos(th3)*(cos(th1)*cos(th2)-sin(th1)*sin(th2)) /(sin(th3)*(sin(th1)
*sin(th2)))) -((sin(th3)*(sin(th1)*cos(th2)+cos(th1)*sin(th2))) /(sin(th3)*(sin
(th1)*sin(th2))))) /(sin(th1+th2+th3)/(sin(th1)*sin(th2)*sin(th3))) by
XCMPLX_1:120
    .= (cot(th3)*((cos(th1)*cos(th2)-sin(th1)*sin(th2))/(sin(th1)*sin(th2)))
-((sin(th3)*(sin(th1)*cos(th2)+cos(th1)*sin(th2))) /(sin(th3)*(sin(th1)*sin(th2
  ))))) /(sin(th1+th2+th3)/(sin(th1)*sin(th2)*sin(th3))) by XCMPLX_1:76
    .= (cot(th3)*((cos(th1)*cos(th2)/(sin(th1)*sin(th2))) -(sin(th1)*sin(th2
  ))/(sin(th1)*sin(th2))) -((sin(th3)*(sin(th1)*cos(th2)+cos(th1)*sin(th2))) /(
sin(th3)*(sin(th1)*sin(th2))))) /(sin(th1+th2+th3)/(sin(th1)*sin(th2)*sin(th3))
  ) by XCMPLX_1:120
    .= (cot(th3)*(cot(th1)*(cos(th2)/(sin(th2))) -(sin(th1)*sin(th2))/(sin(
th1)*sin(th2))) -((sin(th3)*(sin(th1)*cos(th2)+cos(th1)*sin(th2))) /(sin(th3)*(
  sin(th1)*sin(th2))))) /(sin(th1+th2+th3)/(sin(th1)*sin(th2)*sin(th3))) by
XCMPLX_1:76
    .= (cot(th3)*(cot(th1)*cot(th2)-1) -(((sin(th1)*cos(th2)+cos(th1)*sin(
  th2))*sin(th3)) /(sin(th1)*sin(th2)*sin(th3)))) /(sin(th1+th2+th3)/(sin(th1)*
  sin(th2)*sin(th3))) by A1,A2,XCMPLX_1:60
    .= (cot(th3)*(cot(th1)*cot(th2)-1) -((sin(th1)*cos(th2)+cos(th1)*sin(th2
))/(sin(th1)*sin(th2)))) /(sin(th1+th2+th3)/(sin(th1)*sin(th2)*sin(th3))) by A3
,XCMPLX_1:91
    .= (cot(th3)*(cot(th1)*cot(th2)-1) -((sin(th1)*cos(th2)/(sin(th1)*sin(
th2))) +(cos(th1)*sin(th2))/(sin(th1)*sin(th2)))) /(sin(th1+th2+th3)/(sin(th1)*
  sin(th2)*sin(th3))) by XCMPLX_1:62
    .= (cot(th3)*(cot(th1)*cot(th2)-1) -(cos(th2)/sin(th2)+(cos(th1)*sin(th2
))/(sin(th1)*sin(th2)))) /(sin(th1+th2+th3)/(sin(th1)*sin(th2)*sin(th3))) by A1
,XCMPLX_1:91
    .= (cot(th3)*(cot(th1)*cot(th2)-1)-(cot(th2)+cos(th1)/sin(th1))) /(sin(
  th1+th2+th3)/(sin(th1)*sin(th2)*sin(th3))) by A2,XCMPLX_1:91
    .= (cot(th1)*cot(th2)*cot(th3)-cot(th1)-cot(th2)-cot(th3)) /((sin(th1+
th2)*cos(th3)+cos(th1+th2)*sin(th3)) /(sin(th1)*sin(th2)*sin(th3))) by
SIN_COS:75
    .= (cot(th1)*cot(th2)*cot(th3)-cot(th1)-cot(th2)-cot(th3)) /(sin(th1+th2
  )*cos(th3)/(sin(th1)*sin(th2)*sin(th3)) +cos(th1+th2)*sin(th3)/(sin(th1)*sin(
  th2)*sin(th3))) by XCMPLX_1:62
    .= (cot(th1)*cot(th2)*cot(th3)-cot(th1)-cot(th2)-cot(th3)) /(sin(th1+th2
  )*cos(th3)/(sin(th1)*sin(th2)*sin(th3)) +cos(th1+th2)/(sin(th1)*sin(th2)))
by A3,XCMPLX_1:91
    .= (cot(th1)*cot(th2)*cot(th3)-cot(th1)-cot(th2)-cot(th3)) /(cot(th3)*(
  sin(th1+th2)/(sin(th1)*sin(th2))) +cos(th1+th2)/(sin(th1)*sin(th2))) by
XCMPLX_1:76
    .= (cot(th1)*cot(th2)*cot(th3)-cot(th1)-cot(th2)-cot(th3)) /(cot(th3)*((
  sin(th1)*cos(th2)+cos(th1)*sin(th2))/(sin(th1)*sin(th2))) +cos(th1+th2)/(sin(
  th1)*sin(th2))) by SIN_COS:75
    .= (cot(th1)*cot(th2)*cot(th3)-cot(th1)-cot(th2)-cot(th3)) /(cot(th3)*((
sin(th1)*cos(th2)/(sin(th1)*sin(th2)) +cos(th1)*sin(th2)/(sin(th1)*sin(th2))))
  +cos(th1+th2)/(sin(th1)*sin(th2))) by XCMPLX_1:62
    .= (cot(th1)*cot(th2)*cot(th3)-cot(th1)-cot(th2)-cot(th3)) /(cot(th3)*(
cos(th2)/sin(th2)+cos(th1)*sin(th2)/(sin(th1)*sin(th2))) +cos(th1+th2)/(sin(th1
  )*sin(th2))) by A1,XCMPLX_1:91
    .= (cot(th1)*cot(th2)*cot(th3)-cot(th1)-cot(th2)-cot(th3)) /(cot(th3)*(
cot(th2)+cos(th1)/sin(th1)) +cos(th1+th2)/(sin(th1)*sin(th2))) by A2,
XCMPLX_1:91
    .= (cot(th1)*cot(th2)*cot(th3)-cot(th1)-cot(th2)-cot(th3)) /(cot(th3)*(
cot(th2)+cot(th1)) +(cos(th1)*cos(th2)-sin(th1)*sin(th2))/(sin(th1)*sin(th2)))
  by SIN_COS:75
    .= (cot(th1)*cot(th2)*cot(th3)-cot(th1)-cot(th2)-cot(th3)) /(cot(th3)*(
cot(th2)+cot(th1)) +(cos(th1)*cos(th2)/(sin(th1)*sin(th2)) -sin(th1)*sin(th2)/(
  sin(th1)*sin(th2)))) by XCMPLX_1:120
    .= (cot(th1)*cot(th2)*cot(th3)-cot(th1)-cot(th2)-cot(th3)) /(cot(th3)*(
  cot(th2)+cot(th1)) +(cos(th1)*cos(th2)/(sin(th1)*sin(th2))-1)) by A1,A2,
XCMPLX_1:60
    .= (cot(th1)*cot(th2)*cot(th3)-cot(th1)-cot(th2)-cot(th3)) /(cot(th3)*(
  cot(th2)+cot(th1)) +(cot(th1)*(cos(th2)/sin(th2))-1)) by XCMPLX_1:76
    .= (cot(th1)*cot(th2)*cot(th3)-cot(th1)-cot(th2)-cot(th3)) /(cot(th2)*
  cot(th3)+cot(th3)*cot(th1)+cot(th1)*cot(th2)-1);
  hence thesis;
end;
