reserve th, th1, th2, th3 for Real;

theorem
  sin(th1)<>0 & sin(th2)<>0 implies cot(th1+th2) = (cot(th1)*cot(th2)-1)
  /(cot(th2)+cot(th1))
proof
  assume that
A1: sin(th1)<>0 and
A2: sin(th2)<>0;
  cot(th1+th2)=(cos(th1+th2)/(sin(th1)*sin(th2))) /(sin(th1+th2)/(sin(th1)
  *sin(th2))) by A1,A2,XCMPLX_1:55
    .=((cos(th1)*cos(th2)-sin(th1)*sin(th2))/(sin(th1)*sin(th2))) /(sin(th1+
  th2)/(sin(th1)*sin(th2))) by SIN_COS:75
    .=((cos(th1)*cos(th2)-sin(th1)*sin(th2))/(sin(th1)*sin(th2))) /((sin(th1
  )*cos(th2)+cos(th1)*sin(th2))/(sin(th1)*sin(th2))) by SIN_COS:75
    .=(cos(th1)*cos(th2)/(sin(th1)*sin(th2)) -sin(th1)*sin(th2)/(sin(th1)*
  sin(th2))) /((sin(th1)*cos(th2)+cos(th1)*sin(th2))/(sin(th1)*sin(th2))) by
XCMPLX_1:120
    .=(cos(th1)*cos(th2)/(sin(th1)*sin(th2)) -sin(th1)*sin(th2)/(sin(th1)*
  sin(th2))) /(sin(th1)*cos(th2)/(sin(th1)*sin(th2)) +(cos(th1)*sin(th2))/(sin(
  th1)*sin(th2))) by XCMPLX_1:62
    .=(cos(th1)/sin(th1)*(cos(th2)/sin(th2)) -sin(th1)*sin(th2)/(sin(th1)*
  sin(th2))) /(sin(th1)*cos(th2)/(sin(th1)*sin(th2)) +(cos(th1)*sin(th2))/(sin(
  th1)*sin(th2))) by XCMPLX_1:76
    .=(cos(th1)/sin(th1)*(cos(th2)/sin(th2)) -sin(th1)/sin(th1)*(sin(th2)/
  sin(th2))) /(sin(th1)*cos(th2)/(sin(th1)*sin(th2)) +(cos(th1)*sin(th2))/(sin(
  th1)*sin(th2))) by XCMPLX_1:76
    .=(cos(th1)/sin(th1)*(cos(th2)/sin(th2)) -sin(th1)/sin(th1)*(sin(th2)/
sin(th2))) /((sin(th1)/sin(th1))*(cos(th2)/sin(th2)) +(cos(th1)*sin(th2))/(sin(
  th1)*sin(th2))) by XCMPLX_1:76
    .=(cos(th1)/sin(th1)*(cos(th2)/sin(th2)) -sin(th1)/sin(th1)*(sin(th2)/
sin(th2))) /((sin(th1)/sin(th1))*(cos(th2)/sin(th2)) +(cos(th1)/sin(th1))*(sin(
  th2)/sin(th2))) by XCMPLX_1:76
    .=(cos(th1)/sin(th1)*(cos(th2)/sin(th2))-sin(th1)/sin(th1)) /((sin(th1)/
sin(th1))*(cos(th2)/sin(th2)) +(cos(th1)/sin(th1))*(sin(th2)/sin(th2))) by A2,
XCMPLX_1:88
    .=(cos(th1)/sin(th1)*(cos(th2)/sin(th2))-1) /((sin(th1)/sin(th1))*(cos(
  th2)/sin(th2)) +(cos(th1)/sin(th1))*(sin(th2)/sin(th2))) by A1,XCMPLX_1:60
    .=(cos(th1)/sin(th1)*(cos(th2)/sin(th2))-1) /((cos(th2)/sin(th2))+(cos(
  th1)/sin(th1))*(sin(th2)/sin(th2))) by A1,XCMPLX_1:88
    .=(cot(th1)*cot(th2)-1)/(cot(th2)+cot(th1)) by A2,XCMPLX_1:88;
  hence thesis;
end;
