The vectors $\mathrm{OA},\mathrm{OB}$ and $\mathrm{OC}$ are:

${\displaystyle \begin{array}{cc}\multicolumn{1}{c}{\mathrm{OA}}& =(0,0,1)\\ \multicolumn{1}{c}{\mathrm{OB}}& =(\cos u,0,\sin u)\\ \multicolumn{1}{c}{\mathrm{OC}}& =(0,\cos v,\sin v).\end{array}}$

Since these are unit vectors, the angles between the vectors, and hence the lengths of the sides of triangle $\mathrm{ABC}$, are:

${\displaystyle \begin{array}{cc}\multicolumn{1}{c}{a}& ={\cos }^{-1}\mathrm{OB}.\mathrm{OC}={\cos }^{-1}\sin u\sin v\\ \multicolumn{1}{c}{b}& ={\cos }^{-1}\mathrm{OA}.\mathrm{OC}={\cos }^{-1}\sin v\\ \multicolumn{1}{c}{c}& ={\cos }^{-1}\mathrm{OB}.\mathrm{OA}={\cos }^{-1}\sin u.\end{array}}$

Hence

${\displaystyle \cos a=\cos b\cos c.}$