  The pairing symmetry of the newly proposed cobalt high temperature
(high-$T_c$) superconductors formed by vertex shared cation-anion tetrahedral
complexes is studied by the methods of mean field, random phase approximation
(RPA) and functional renormalization group (FRG) analysis. The results of all
these methods show that the $d_{x^2-y^2}$ pairing symmetry is robustly favored
near half filling. The RPA and FRG methods, which are valid in weak interaction
regions, predict that the superconducting state is also strongly orbital
selective, namely the $d_{x^2-y^2}$ orbital that has the largest density near
half filling among the three $t_{2g}$ orbitals dominates superconducting
pairing. These results suggest that the new materials, if synthesized, can
provide indisputable test to high-$T_c$ pairing mechanism and the validity of
different theoretical methods.
