Features of low dimensional magnetism resulting from a square-net arrangement of Co atoms in trirutile CoTa$_2$O$_6$ is studied in the present work by means of density functional theory and is compared with the experimental results of specific heat and neutron diffraction. The small total energy differences between the ferromagnetic (FM) and antiferromagnetic (AFM) configuration of CoTa$_2$O$_6$ shows that competing magnetic ground states exist, with the possibility of transition from FM to AFM phase at low temperature. Our calculation further suggests the semi-conducting behavior for CoTa$_2$O$_6$ with a band gap of $sim$0.41 eV. The calculated magnetic anisotropy energy is $sim$2.5 meV with its easy axis along the [100] (in-plane) direction. Studying the evolution of magnetism in Co$_{1-x}$Mg$_x$Ta$_2$O$_6$ (x = 0, 0.1, 0.3, 0.5, 0.7 and 1). it is found that the sharp AFM transition exhibited by CoTa$_2$O$_6$ at $T_N$ = 6.2 K in its heat capacity vanishes with Mg-dilution, indicating the obvious effect of weakening the superexchange pathways of Co. The current specific heat study reveals the robust nature of $T_N$ for CoTa$_2$O$_6$ in applied magnetic fields. Clear indication of short-range magnetism is obtained from the magnetic entropy, however, diffuse components are absent in neutron diffraction data. At $T_N$, CoTa$_2$O$_6$ enters a long-range ordered magnetic state which can be described using a propagation vector, (1/4, 1/4, 0). Upon Mg-dilution at $x geq$0.1, the long-range ordered magnetism is destroyed. The present results should motivate an investigation of magnetic excitations in this low-dimensional anisotropic magnet.
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