How can we show that $\operatorname{Tor}_1^A(\hat{A},G_f/G)=0$?

Question

This question is quite closely related to my last question:

Is it always true that $\hat{A} \otimes_A A_f/A \cong \hat{A}_f/ \hat{A}$?

Let $A$ a commutative ring, $f \in A$ a non zerodivisor. Let $\hat{A}$ the $f$-adic completion of $A$. Let $G$ be an $\hat{A}$-Module without $f$-torsion.

How can we show that $\operatorname{Tor}_1^A(\hat{A},G_f/G)=0$?

Edit: of course (as in my linked question), it holds again that $\varinjlim G/f^n G = G_f/G$, therefore it would suffice to show $\operatorname{Tor}_1^A(\hat{A}, G/f^n G) =0$ for all $n \geq 0$.