Let $S_g$ denote the closed orientable surface of genus $g\geq 2$. Then there is a natural bijection between the Teichmuller space $\text{Teich}(S_g)$ of $S_g$ and the set of all the discrete-faithful representations (up to conjugation) $\text{DF}(\pi_1S_g, \text{PSL}_2(\mathbf R))/\text{PSL}_2(\mathbf R)$.
On pg. 284 of Farb and Margalit's A Primer on Mapping Class Groups, the following is mentioned:
Let $\gamma\in \pi_1(S_g)$. Then for $\mathcal X\in \text{Teich}(S_g)$, the length $l_{\mathcal X}(\gamma)$ of the geodesic representative of $\gamma$ is same as $2\cosh^{-1}(\text{trace}(\rho_{\mathcal X}(\gamma))/2)$, where $\rho_{\mathcal X}$ is the discrete faithful representation corresponding to $\mathcal X$.
I am unable to see how this formula comes about. Can somebody please provide a proof or a reference? Thanks.