Construct a homotopy of geodesics from "common perpendicular" $h$ to a geodesic $c'$ from $c(0)$ to $c(1)$  with the ends are on A and B.
(This is possible since curvature is negative.)
In this process, the number of self intersections stays constant;
i.e. $c'$ is simple.
Thus, we get two geodesics $c$ and $c'$ with common ends.
Since curvature is negative,  $c\sim c'$ implies $c=c'$, which is not possible if $c$ is non-simple.