  We investigate the rare decays $B_c \rightarrow D_s(1968) \ell \bar{\ell}$
and $B_c\rightarrow D_s^*(2317) \ell \bar{\ell}$ in the framework of the
light-cone quark model (LCQM). The transition form factors are calculated in
the space-like region and then analytically continued to the time-like region
via exponential parametrization. The branching ratios and longitudinal lepton
polarization asymmetries (LPAs) for the two decays are given and compared with
each other. The results are helpful to investigating the structure of $B_c$
meson and to testing the unitarity of CKM quark mixing matrix. All these
results can be tested in the future experiments at the LHC.
