In this paper, we consider a cooperative rate-splitting (CRS) scenario in a terahertz (THz) downlink system. Specifically, we introduce a new CRS framework called extraction-based CRS (eCRS) that exploits the benefits of rate-splitting, grouping, and data extraction. Furthermore, by considering characteristics of the THz communication systems, we construct a novel cooperative channel model. We examine an extreme case of eCRS, where an optimization problem is formulated and solved based on the cooperative channel model. In simulation results, we corroborate the effectiveness of our proposed framework and the impact of the cooperative channel model.