reserve th, th1, th2, th3 for Real;

theorem
  cos(th1)<>0 & cos(th2)<>0 implies sin(th1+th2)/sin(th1-th2)= (tan(th1)
  +tan(th2))/(tan(th1)-tan(th2))
proof
  assume that
A1: cos(th1)<>0 and
A2: cos(th2)<>0;
  sin(th1+th2)/sin(th1-th2) = (sin(th1)*cos(th2)+cos(th1)*sin(th2))/sin(
  th1-th2) by SIN_COS:75
    .= (sin(th1)*cos(th2)+cos(th1)*sin(th2)) /(sin(th1)*cos(th2)-cos(th1)*
  sin(th2)) by SIN_COS:82
    .= ((sin(th1)*cos(th2)+cos(th1)*sin(th2))/(cos(th1)*cos(th2))) /((sin(
  th1)*cos(th2)-cos(th1)*sin(th2))/(cos(th1)*cos(th2))) by A1,A2,XCMPLX_1:55
    .= ((sin(th1)*cos(th2)/(cos(th1)*cos(th2))) +cos(th1)*sin(th2)/(cos(th1)
*cos(th2))) /((sin(th1)*cos(th2)+(-cos(th1)*sin(th2)))/(cos(th1)*cos(th2))) by
XCMPLX_1:62
    .= ((sin(th1)*cos(th2)/(cos(th1)*cos(th2))) +cos(th1)*sin(th2)/(cos(th1)
  *cos(th2))) /((sin(th1)*cos(th2)/(cos(th1)*cos(th2))) +(-cos(th1)*sin(th2))/(
  cos(th1)*cos(th2))) by XCMPLX_1:62
    .= (((sin(th1)/cos(th1))*(cos(th2)/cos(th2))) +cos(th1)*sin(th2)/(cos(
th1)*cos(th2))) /((sin(th1)*cos(th2)/(cos(th1)*cos(th2))) +(-cos(th1)*sin(th2))
  /(cos(th1)*cos(th2))) by XCMPLX_1:76
    .= (((sin(th1)/cos(th1))*(cos(th2)/cos(th2))) +(cos(th1)/cos(th1))*(sin(
th2)/cos(th2))) /((sin(th1)*cos(th2)/(cos(th1)*cos(th2))) +(-cos(th1)*sin(th2))
  /(cos(th1)*cos(th2))) by XCMPLX_1:76
    .= (((sin(th1)/cos(th1))*(cos(th2)/cos(th2))) +(cos(th1)/cos(th1))*(sin(
th2)/cos(th2))) /((sin(th1)/cos(th1)*(cos(th2)/cos(th2))) +(cos(th1))*(-sin(th2
  ))/(cos(th1)*cos(th2))) by XCMPLX_1:76
    .= (((sin(th1)/cos(th1))*(cos(th2)/cos(th2))) +(cos(th1)/cos(th1))*(sin(
th2)/cos(th2))) /((sin(th1)/cos(th1)*(cos(th2)/cos(th2))) +(cos(th1)/cos(th1)*(
  (-sin(th2))/cos(th2)))) by XCMPLX_1:76
    .= (sin(th1)/cos(th1)+(cos(th1)/cos(th1))*(sin(th2)/cos(th2))) /((sin(
th1)/cos(th1)*(cos(th2)/cos(th2))) +(cos(th1)/cos(th1)*((-sin(th2))/cos(th2))))
  by A2,XCMPLX_1:88
    .= (sin(th1)/cos(th1)+(sin(th2)/cos(th2))) /((sin(th1)/cos(th1)*(cos(th2
  )/cos(th2))) +(cos(th1)/cos(th1)*((-sin(th2))/cos(th2)))) by A1,XCMPLX_1:88
    .= (sin(th1)/cos(th1)+(sin(th2)/cos(th2))) /(sin(th1)/cos(th1)+(cos(th1)
  /cos(th1)*((-sin(th2))/cos(th2)))) by A2,XCMPLX_1:88
    .= (sin(th1)/cos(th1)+(sin(th2)/cos(th2))) /(sin(th1)/cos(th1)+(-sin(th2
  ))/cos(th2)) by A1,XCMPLX_1:88
    .= (tan(th1)+tan(th2))/(tan(th1)+-sin(th2)/cos(th2)) by XCMPLX_1:187
    .= (tan(th1)+tan(th2))/(tan(th1)-tan(th2));
  hence thesis;
end;
