  Inspired by the anomalies in $B\to D^{(*)} \tau \bar{\nu}_\tau$ decay, in
this work, we carefully study the influence of the most general dimension-six
effective operators involving right-handed neutrinos on the four cascade decays
$B_c \to \tau(\to \nu_\tau h ) \bar{\nu}_\tau $ ($h=\pi,\rho$) and $B_c \to
\tau(\to \nu_\tau \ell \bar{\nu}_\ell) \bar{\nu}_\tau $ ($\ell =\mu, e$). We
calculate for the first time the analytical results of the distributions of the
differential decay rates of these cascade decays with respect to the energy of
the charged particles in the final state. We select a total of 13 new physics
benchmark points, among which 9 are derived from the effect of purely
left-handed neutrinos, and the other 4 are from the effect of purely
right-handed neutrinos. We find that the benchmark points BP4, BP6, BP9, BP11,
and BP13 can significantly increase the values of these distributions while the
BP3 causes a sharp decrease in these distributions. The end-point behavior of
the differential distribution $d\Gamma/dE_\pi$ can be used to disentangle the
effects of left-handed and right-handed neutrinos. Finally, in order to
eliminate the uncertainties brought by the CKM matrix element $V_{cb}$ and the
decay constant $f_{B_c}$, we introduce the normalized distributions
$d\Gamma/(\Gamma dE_a)$ ($a=\pi,\rho,\mu,e$), which are only sensitive to the
right-handed neutrinos. We find that in each normalized distribution there
exists a fixed point that is not related to any dimension-six effective
operators considered in this work.
