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We prove the identity $$\prod_{\gamma}\left(\frac{e^{l(\gamma)}+1}{e^{l(\gamma)}-1}\right)^{2h}=\exp\left(\frac{l_1+l_2+l_3}{2}\right),$$ (or $$\prod_{\gamma}\left(\frac{t(\gamma)^2}{t(\gamma)^2-4}\right)^h=\frac{t_1+\sqrt{t_1^2-4}}{2}\cdot\frac{t_2+\sqrt{t_2^2-4}}{2}\cdot\frac{t_3+\sqrt{t_3^2-4}}{2}$$ in trace coordinates), where the product is over all simple closed geodesicson the once-punctured torus, $l(\gamma)=2\operatorname{arccosh}(t(\gamma)/2)$is the length of the geodesic, and $l_i$ ($t_i$) are the lengths (traces) ofany triple of simple geodesics $\{\gamma_i\}$ intersecting at a single point.The exponent $h=h(\gamma;\{\gamma_i\})$ is a positive integer "height" whichincreases as we move away from the chosen triple $\{\gamma_i\}$ in its orbitunder $SL_2(\mathbb{Z})$ (see Figure 1 for the "definition by picture"). Forcomparison, a short proof of McShane's identity $$\sum_{\gamma}\frac{1}{1+e^{l(\gamma)}}=\frac{1}{2}=\sum_{\gamma}\frac{1-\sqrt{1-4/t(\gamma)^2}}{2}$$ in the same spirit is given in an appendix. Both proofs are elementary andproceed by "integrating" around the chosen triple $\{\gamma_i\}$ in itsTeichm\"{u}ller orbit.

An infinite product on the Teichm\"{u}ller space of the once-punctured torus