When an energetic parton traverses the hot QCD medium, it may suffer from multiple scattering and lose its energy. The medium-induced gluon radiation for a massive quark will be suppressed relative to that of a light quark due to the dead-cone effect. The development of new declustering techniques of jet evolution makes a direct study of the dead-cone effect in the QCD medium possible for the first time. In this work, we compute the emission angle distribution of the charm-quark-initiated splittings in $\textit{D}$ $^{0}$ meson tagged jet and that of the light parton-initiated splittings in an inclusive jet in $\textit{p}$+$\textit{p}$ and Pb+Pb at 5.02 TeV by utilizing the declustering techniques of jet evolution. The heavy quark propagation and induced energy loss in the QCD medium are simulated with the SHELL model based on the Langevin equation. By directly comparing the emission angle distributions of charm-quark-initiated splittings with those of light parton-initiated splittings at the same energy intervals of the initial parton, we provide insights into the fundamental splitting structure in $\textit{A}$+$\textit{A}$ collisions, thereby exploring the possible observation of the dead-cone effect in medium-induced radiation. We further investigate the case of the emission angle distributions normalized to the number of splittings and find the dead-cone effect will broaden the emission angle of the splitting and reduce the possibility for such splitting to occur, leading the massive parton to lose less energy. We also find that the collisional energy loss mechanism has a negligible impact on the medium modification to the emission angle distribution of the charm-quark-initiated splittings for $\textit{D}$ $^{0 }$meson-tagged jets.