reserve th, th1, th2, th3 for Real;

theorem
  cos(th1) <> 0 & cos(th2) <> 0 implies cos(th1+th2)/cos(th1-th2) = (1-
  tan(th1)*tan(th2))/(1+tan(th1)*tan(th2))
proof
  assume
A1: cos(th1) <>0 & cos(th2) <> 0;
  cos(th1+th2)/cos(th1-th2) =(cos(th1)*cos(th2)-sin(th1)*sin(th2))/cos(th1
  -th2) by SIN_COS:75
    .=((cos(th1)*cos(th2)-sin(th1)*sin(th2))/(cos(th1)*cos(th2))) /(cos(th1-
  th2)/(cos(th1)*cos(th2))) by A1,XCMPLX_1:55
    .=(cos(th1)*cos(th2)/(cos(th1)*cos(th2)) -(sin(th1)*sin(th2))/(cos(th1)*
  cos(th2))) /(cos(th1-th2)/(cos(th1)*cos(th2))) by XCMPLX_1:120
    .=(1-(sin(th1)*sin(th2))/(cos(th1)*cos(th2))) /(cos(th1-th2)/(cos(th1)*
  cos(th2))) by A1,XCMPLX_1:60
    .=(1-tan(th1)*(sin(th2)/cos(th2))) /(cos(th1-th2)/(cos(th1)*cos(th2)))
  by XCMPLX_1:76
    .=(1-tan(th1)*tan(th2)) /((cos(th1)*cos(th2)+sin(th1)*sin(th2))/(cos(th1
  )*cos(th2))) by SIN_COS:83
    .=(1-tan(th1)*tan(th2)) /(cos(th1)*cos(th2)/(cos(th1)*cos(th2)) +sin(th1
  )*sin(th2)/(cos(th1)*cos(th2))) by XCMPLX_1:62
    .=(1-tan(th1)*tan(th2)) /(1+sin(th1)*sin(th2)/(cos(th1)*cos(th2))) by A1,
XCMPLX_1:60
    .=(1-tan(th1)*tan(th2))/(1+tan(th1)*tan(th2)) by XCMPLX_1:76;
  hence thesis;
end;
