Suppose that $f \colon U \to V$ is a diffeomorphism of planar domains. Its differential $Df$ can be pointwise expressed the sum of a complex-linear and complex-anti-linear mapping: Given a tangent vector $\xi$, $$Df\xi = f_z \xi + f_{\overline{z}}\overline{\xi}.$$ If $f$ is orientation preserving, then $|f_z|>|f_{\overline{z}}|$ and the Beltrami coefficient of $f$ is the defined to be the complex-valued function $$\mu_f:= \frac{f_z}{f_{\overline{z}}}.$$
The diffeomorphism $f$ defines a Riemannian metric on $U$ via the pullback of the usual Euclidean inner product on $V$: given tangent vectors $\xi$ and $\zeta$, $$\langle \xi, \zeta \rangle_f := \langle Df\xi, Df\zeta \rangle_{\text{Euc}} = \xi^T(Df^TDf)\zeta.$$
Note that the entries of the positive symmetric matrix $Df^TDf$ determine the Beltrami coefficient of $\mu$: if $Df^TDf=\begin{bmatrix}E & F \\ F & G\\ \end{bmatrix}$, then $$\mu_f = \frac{E - G + 2iF}{E+G+2\sqrt{EG-F^2}}.$$
Now, suppose that a Riemannian metric $g$ on $U$ is given, and its expression as a matrix is given by $\begin{bmatrix}E_g & F_g \\ F_g & G_g\\ \end{bmatrix}$. Moreover, suppose that we have an orientation preserving diffeomorphism $f \colon U \to V$ so that $$\mu_f = \frac{E_g- G_g + 2iF_g}{E_g+G_g+2\sqrt{E_gG_g-F_g^2}} =:\mu_g.$$
My understanding is that the pullback metric $\langle \cdot, \cdot \rangle_f$ is conformally equivalent to $g$, but I am unable to verify this. This is essentially stated as obvious on the wikipedia pages
https://en.wikipedia.org/wiki/Beltrami_equation
https://en.wikipedia.org/wiki/Isothermal_coordinates
https://en.wikipedia.org/wiki/Quasiconformal_mapping
but I am unable to follow the reasoning.
My attempt at a proof was as follows. Since $\mu_f =\mu_g$, we may write $$Df\xi = f_z(\xi + \mu_g\overline{\xi}).$$ Writing $W\xi := \mu_g\overline{\xi}$, we may then note that as $W$ is symmetric, $$Df^TDf = |f_z|^2(\text{Id} + W)^2.$$ We may calculate $$(\text{Id} + W)^2 = \begin{bmatrix} (1+\text{Re}\ \mu_g)^2 + (\text{Im}\ \mu_g)^2 & 2\text{Im}\ \mu_g \\2\text{Im}\ \mu_g &(1-\text{Re}\ \mu_g)^2 + (\text{Im}\ \mu_g)^2\end{bmatrix}.$$
My hope was then to use the definition of $\mu_g$, plugging in the real and imaginary parts, and then see the relationship between $Df^TDf$ and $g$. Unfortunately, my calculations just lead to a mess and don't simplify.
The key point I am missing is the following: how exactly do the complex number $\mu_g$ and the real number $|f_z|^2$ determine the three real numbers $E_g, F_g, G_g$?
Thanks for reading.