Transformation Poincaré-coordinates to global coordinates in $\mathrm{AdS}_2$ For the two dimensional anti de-sitter space $\mathrm{AdS}_2$ one can consider the Poincaré-coordinates $\mathrm{d}s^2_P = -r^2 \mathrm{d}t^2 + \frac{1}{r^2} \mathrm{d}r^2$ which covers only half of the spacetime. Most of the time though one uses the global coordinates $\mathrm{d}s^2_{G1} = -(1+r^2) \mathrm{d}t^2 + \frac{1}{1+r^2} \mathrm{d}r^2$, or the latter with a additional transformation $r = \mathrm{sinh}(\rho)$: $\mathrm{d}s^2_{G2} = -\mathrm{cosh}^2(\rho) \mathrm{d}t^2 + \mathrm{d}\rho^2$. This question seems to be more in the trivial side, but what is the explicit coordinate transformation from $\mathrm{d}s^2_P$ to either $\mathrm{d}s^2_{G1}$ or $\mathrm{d}s^2_{G2}$?
 A: The coordinate transformation from $(r,t)$ to $(\rho,\tau)$ given by
$$\rho= \cosh r \cos t+\sinh r,\;\;\tau= \frac{\cosh r \sin t }{\cosh ^2 r \sin ^2 t-1}(\sinh r-\cosh r \cos t),$$
$$\text{in the range}\;\;r>0, \;-\pi/2<t<\pi/2\Leftrightarrow\rho>0,\;-\infty<\tau<\infty,$$
converts the global metric
$$ds^2 = - \cosh^2 r \,dt^2 + dr^2,$$
into the Poincaré metric
$$ds^2 =  - \rho^2 d\tau^2+\rho^{-2} d\rho^2.$$
I searched the literature but could not find this coordinate transformation in the explicit form requested by the OP. I have verified its correctness algebraically: 
$$d\rho=-dt \sin t\cosh r+ dr(\cosh r+ \sinh r \cos t),\;\;d\tau=\frac{dt(\sinh 2 r \cos t+2  \cosh ^2 r)-2 dr \sin t}{2 (\cosh r \cos t+\sinh r)^2}$$
$$\Rightarrow - \rho^2 d\tau^2+\rho^{-2} d\rho^2=- \cosh^2 r \,dt^2 + dr^2,\;\;\text{as desired}.$$

A: See chapter 2 of these lecture notes: https://sites.krieger.jhu.edu/jared-kaplan/files/2016/05/AdSCFTCourseNotesCurrentPublic.pdf
All the relevant coordinate changes for $AdS_d$ are in there, and the specialization to $d=2$ is immediate.
