It is well known that the class of trivial ring extensions covers Morita context rings with zero bimodule homomorphisms, particularly, covers formal triangular matrix rings. In this talk, we first give some homological formulas over trivial ring extensions. Then we investigate how to construct (hereditary, complete) cotorsion pairs and (hereditary, cofibrantly generated) Hovey triples over trivial ring extensions. Finally, some applications are given in some Morita context rings.