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Numerous studies claim that terahertz (THz) communication will be an essential piece of sixth-generation wireless communication systems. Its promising potential also comes with major challenges, in particular the reduced coverage due to harsh propagation loss, hardware constraints, and blockage vulnerability. To increase the coverage of THz communication, we revisit cooperative communication. We propose a new type of cooperative rate-splitting (CRS) called extraction-based CRS (eCRS). Furthermore, we explore two extreme cases of eCRS, namely, identical eCRS and distinct eCRS. To enable the proposed eCRS framework, we design a novel THz cooperative channel model by considering unique characteristics of THz communication. Through mathematical derivations and convex optimization techniques considering the THz cooperative channel model, we derive local optimal solutions for the two cases of eCRS and a global optimal closed form solution for a specific scenario. Finally, we propose a novel channel estimation technique that not only specifies the channel value, but also the time delay of the channel from each cooperating user equipment to fully utilize the THz cooperative channel. In simulation results, we verify the validity of the two cases of our proposed framework and channel estimation technique.