Quarter-wave impedance transformer working principle
Zin = (RL)^2/Z0


No, that is incorrect.

When you have a quarter wave transformer the \$\tan(\beta\ell)\$ part goes to infinity hence the formula for input impedance becomes: -

$$Z_{IN} = Z_0\cdot\dfrac{Z_0}{R_L}$$

or

$$Z_{IN} = Z_0^2\dfrac{1}{R_L}$$

Hence an open circuit terminating the line at the load end produces a short at the input end and, a short at the load end produces an open circuit at the input. If RL = Z0 then Zin = Z0.