In this article, we first introduce new Frenet formulas by making use of the properties of connected curves. Then applying these formulas we show that some decisive properties of Bertrand partner curve can be given in terms of a Bertrand curve. More precisely, we offer differential equations of the Bertrand partner curve with respect to both Levi-Civita and normal Levi-Civita connections in terms of the Bertrand curve. We also give harmonicity conditions of the partner curve of Bertrand curve pair by the same method. We obtain some new results and finally we give an example to support our allegations.