  In $B_c^- \to J/\psi (\to \mu^+ \mu^-)\tau^-\bar{\nu}_\tau$ decay, the
three-momentum $\boldsymbol{p}_{\tau^-}$ cannot be determined accurately due to
the decay products of $\tau^-$ inevitably include an undetected $\nu_{\tau}$.
As a consequence, the angular distribution of this decay cannot be measured. In
this work, we construct a {\it measurable} angular distribution by considering
the subsequent decay $\tau^- \to \pi^- \nu_\tau$. The full cascade decay is
$B_c^- \to J/\psi (\to \mu^+ \mu^-)\tau^- (\to \pi^- \nu_\tau)\bar{\nu}_\tau$,
in which the three-momenta $\boldsymbol{p}_{\mu^+}$, $\boldsymbol{p}_{\mu^-}$,
and $\boldsymbol{p}_{\pi^-}$ can be measured. The five-fold differential
angular distribution containing all Lorentz structures of the new physics (NP)
effective operators can be written in terms of twelve angular observables
$\mathcal{I}_i (q^2, E_\pi)$. Integrating over the energy of pion $E_\pi$, we
construct twelve normalized angular observables $\widehat{\mathcal{I}}_i(q^2)$
and two lepton-flavor-universality ratios $R(P_{L,T}^{J/\psi})(q^2)$. Based on
the $B_c \to J/\psi$ form factors calculated by the latest lattice QCD and sum
rule, we predict the $q^2$ distribution of all $\widehat{\mathcal{I}}_i$ and
$R(P_{L,T}^{J/\psi})$ both within the Standard Model and in eight NP benchmark
points. We find that the benchmark BP2 (corresponding to the hypothesis of
tensor operator) has the greatest effect on all $\widehat{\mathcal{I}}_{i}$ and
$R(P_{L,T}^{J/\psi})$, except $\widehat{\mathcal{I}}_{5}$. The ratios
$R(P_{L,T}^{J/\psi})$ are more sensitive to the NP with pseudo-scalar operators
than the $\widehat{\mathcal{I}}_{i}$. Finally, we discuss the symmetries in the
angular observables and present a model-independent method to determine the
existence of tensor operators.
