general relativity - Geodesic distance in de Sitter space

Consider $N$ dimensional de Sitter space embedded in $N+1$ dimensional Minkowski space:

$$\eta_{\mu\nu}X^\mu X^\nu=1, \hspace{1cm}\eta_{\mu\nu}=\text{diag}(-1,1,\dots,1)$$ where I set $H=1$ for simplicity. Given two points in de Sitter space (denoting de Sitter coordinates by lower case $x$, as opposed to capitalized Minkowski coordinates $X$) we define the so-called hyperbolic distance $$P(x,x')=g_{\mu\nu}x^\mu x'^\nu$$ It is often (between eqn. 32 and 33, between eqn. 2.2 and 2.3, equation 3) said that this is related to the geodesic distance $$D(x,x')=\int_x^{x'}d\lambda\sqrt{g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx'^\nu}{d\lambda}}$$ by $P=\cos D$. This is independent of the specific coordinates one chooses on de Sitter space. However, I cannot seem to understand how to show this. Can anyone here perhaps enlighten me? Any help would be much appreciated.