 Solution

A great-circle arc of angle $\arccos (p\cdot q)\le \pi$ joins the points. The spherical distance inequality follows by integrating the polar metric $d{s}^2=d{r}^2+{\sin }^{2}r,d{\theta }^2$, so every curve has length at least the radial change. If $q\ne -p$, the shorter great-circle arc is unique; equality forces constant angular coordinate and monotone radial motion, hence only a reparametrisation.