Calculating the topological cyclic homology (TC) of a trivial square zero extension of ring spectra is a key input in the proof of the Dundas-Goodwillie-McCarthy theorem, one of the most important results in trace methods to date. TC is built from topological Hochschild homology (THH) by making use of the cyclotomic structure on THH, and in the case of a trivial square zero extension of a ring spectrum by a bimodule, the cyclotomic structure is completely determined by cyclic versions of THH with coefficients. In this talk, I will give a brief overview of trace methods, define these cyclic versions of THH with coefficients, and explain how to obtain the cyclotomic structure on THH of a trivial square zero extension. Time permitting, I will discuss some applications to topological restriction homology.