The distributions and abundances of molecules in protoplanetary disks are powerful tracers of the physical and chemical disk structures. The abundance ratios of HCN and its isomer HNC are known to be sensitive to gas temperature. Their line ratios might therefore offer a unique opportunity to probe the properties of the emitting gas. Using the 2D thermochemical code DALI, we ran a set of disk models accounting for different stellar and disk properties, with an updated chemical network for the nitrogen chemistry. These modeling results were then compared with observations, including new ALMA observations of HNC $J=3-2$ for the TW Hya disk and HNC $J=1-0$ for 29 disks in Lupus. Similar to CN, HCN and HNC have brighter line emission in models with larger disk flaring angles and higher UV fluxes. HNC and HCN are predicted to be abundant in the warm surface layer and outer midplane region, which results in ring-shaped emission patterns. However, the precise emitting regions and emission morphology depend on the probed transition, as well as on other parameters such as C and O abundances. The modeled HNC-to-HCN line intensity ratio increases from in the inner disk to up to 0.8 in the outer disk regions, which can be explained by efficient HNC destruction at high temperatures. Disk-integrated HNC line fluxes from current scarce observations and its radial distribution in the TW Hya disk are broadly consistent with our model predictions. The HNC-to-HCN flux ratio robustly increases with radius (decreasing temperature), but its use as a chemical thermometer in disks is affected by other factors, including UV flux and C and O abundances. High-spatial resolution ALMA disk observations of HNC and HCN that can locate the emitting layers would have the great potential to constrain both the disk thermal and UV radiation structures, and also to verify our understanding of the nitrogen chemistry.