Solution
= 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 $ds^2=dr^2+\sin^2r,d\theta^2$, 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.