reserve x,x1,x2,x3 for Real;

theorem Th13:
  cos(x)<>0 & sin(x)<>0 implies sec(2*x) = (sec(x))^2/(1-(tan(x))
  ^2) & sec(2*x)=(cot(x)+tan(x))/(cot(x)-tan(x))
proof
  assume that
A1: cos(x)<>0 and
A2: sin(x)<>0;
A3: sec(2*x)=1/((1-(tan(x))^2)/(1+(tan(x))^2)) by A1,Th8
    .=(1+(tan(x))^2)/(1-(tan(x))^2) by XCMPLX_1:57
    .=((sec(x))^2)/(1-(tan(x))^2) by A1,Th11;
  then sec(2*x)=(1+(tan(x))^2)/(1-(tan(x))^2) by A1,Th11
    .=((1+(tan(x))^2)/tan(x))/((1-(tan(x))^2)/tan(x)) by A1,A2,XCMPLX_1:50,55
    .=(1/tan(x)+(tan(x))^2/tan(x))/((1-(tan(x))^2)/tan(x)) by XCMPLX_1:62
    .=(cot(x)+(tan(x))^2/tan(x))/((1-(tan(x))^2)/tan(x)) by XCMPLX_1:57
    .=(cot(x)+(tan(x))^2/tan(x))/(1/tan(x)-(tan(x))^2/tan(x)) by XCMPLX_1:120
    .=(cot(x)+tan(x)*tan(x)/tan(x))/(cot(x)-(tan(x))^2/tan(x)) by XCMPLX_1:57
    .=(cot(x)+tan(x))/(cot(x)-tan(x)*tan(x)/tan(x)) by A1,A2,XCMPLX_1:50,89;
  hence thesis by A1,A2,A3,XCMPLX_1:50,89;
end;
