How to prove it mathematically that either rhombic or the monoclinic sulfur is denser?
$$\frac{\mathrm dP}{\mathrm dT} = \frac{\Delta_\mathrm{trs}S}{\Delta_\mathrm{trs}V}$$

$\Delta_\mathrm{trs}S$ is positive. $\displaystyle\frac{\mathrm dP}{\mathrm dT}$ is said to be positive as well. Thus, $\Delta_\mathrm{trs}V$ has to be positive, i.e. monoclinic sulfur has a greater molar volume, and thus rhombic sulfur is denser.