We would need:

1 - t^2 = r^2 Eqn 1 => rational triplet {x,y,r} = {r,t,1}
1 + t^2 = q^2 Eqn 2 => rational triplet {x,y,r} = {1,t,q}

Substituting for t^2 in Eqn 2:

1 + (1 - r^2) = q^2
2 - r^2 = q^2 => IRRATIONAL triplet {x,y,r} = {r;q;sqrt2}

Contradiction. Therefore the two rational triplets assumed in Eqn 1 and Eqn 2 are impossible. And so sqrt<(1-t^2)/(1+t^2)> is irrational for rational t. (0<t<1)

Does this work ?

I disagree with

> We would need.