Consider the hyperbolic plane $\mathbb{H}$ of dimension $2$ (think of it as the half-plane model). Let $c$ be a geodesic on $\mathbb{H}$ and let $z_1, z_2$ points such that $r=d(z_1,c)=d(z_2,c)$ and let $c(s_1), c(s_2)$ ($s_1< s_2$) be the corresponding foot-point projection on $c$. Also assume that $z_1,z_2$ lies on the same side of the geodesic. Using the formula for the distance between two points and some hyperbolic trigonometry, it is possible to show \begin{equation} \cosh(d(z_1,z_2)/2)= \sinh(r)\cosh((s_2-s_1)/2). \end{equation}
Let now $M$ be a complete, simply connected manifold of dimension $2$ with sectional curvature $K\leq -1$. Consider $\gamma$ a geodesic in $M$, $p_1,p_2\in M$ such that $d(p_i,\gamma)=r$ for $i=1,2$ and assume they lie on the same side of the $\gamma$. Let $\gamma(s_1), \gamma(s_2)$ be the corresponding foot-point projection. I think that the following inequality is true:
\begin{equation} \cosh(d_M(p_1,p_2)/2)\geq \sinh(r)\cosh((s_2-s_1)/2). \end{equation}
However, I cannot find any reference for it and I am not so sure about how to prove it. My idea would be to consider appropriate points $p\in M$, $z\in\mathbb{H}$ and an isometry $i:T_p M\rightarrow T_z \mathbb{H}$ together with the map $\varphi = \exp_z \circ i \circ \exp_p^{-1}$ ($\exp_*$ is the exponential map) and claim that we can choose $i$ so that $d_M(\gamma(s_1), \gamma(s_2))= d_{\mathbb{H}}(\varphi(\gamma(s_1)), \varphi(\gamma(s_2)))$, $d_M(\gamma(s_i), p_i)=d_{\mathbb{H}}(\varphi(\gamma(s_i)), \varphi(p_i))$ for $i=1,2$ and apply the corollary of Rauch's comparison theorem about lengths of curves (see for example Proposition 2.5 in Do Carmo's "Riemannian geometry") to claim $d_M(p_2,p_1)\geq d_{\mathbb{H}}(\varphi(p_2), \varphi(p_1))$ and then use the result for the hyperbolic space.
However, I think that this claim for $i$ may not be true. Any suggestions?
Thanks for the help!