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Proximity coupling of ferromagnetic insulator EuS to the topological insulator Bi$_2$Se$_3$ has been proposed to break time-reversal symmetry near the surface of Bi$_2$Se$_3$, introducing an energy gap or a tilt in the surface Dirac cone. As an inverse proximity effect, strong spin-orbit coupling available in the topological surface states can enhance the Curie temperature of ferromagnetism in EuS largely beyond its bulk value, and also generate a magnetic anisotropy. This can result in a canting of the magnetic moment of Eu ions in a plane perpendicular to the interface. Here, we investigate theoretically electronic transport properties arising from the Bi$_2$Se$_3$/EuS interfaces in the planar Hall geometry. Our analysis, based on a realistic model Hamiltonian and a semi-classical formalism for the Boltzmann transport equation, reveals distinct intriguing features of anisotropic planar Hall conductivity, depending on different scenarios for the canting of the Eu moments: fixed Eu moment canting, and freely-orientable Eu moment in response to the external in-plane magnetic field. The anisotropy in the planar Hall conductivity arises from the asymmetric Berry curvature of the gapped topological surface states. We also explore topological Hall effect of the Dirac surface states, coupled to a skyrmion crystal which can emerge in the EuS due to the interplay of ferromagnetic Heisenberg exchange, interfacial Dzyaloshinskii-Moriya interaction, and perpendicular alignment of the Eu moment. Our study provides new impetus for probing complex interplay between magnetic exchange interactions and topological surface states via anisotropic planar Hall effects., Comment: 13 pages, 7 figures