Relationsip between two definitions of the christoffel symbol?

Question

When I first started learing about tensor calculus, the professor defined the Christoffel symbol as $$\Gamma ^a _{bc} = Z^a \cdot \partial_b Z_c $$ Where $Z^a$ is a contravariant basis vector and $Z_c$ is a covariant basis vector. In general relativity and most contexts as far as I can tell: the Christoffel symbol is $$\Gamma ^a_ {bc} = \frac{1}{2} g^{ad}(\partial _b g_{cd} + \partial _c g_{bd} - \partial_d g_{bc})$$ Where $g^{ab}$ is the contravariant metric tensor and $g_{ab}$ is the covariant metric tensor. I was just wondering what the proof is for the relationship between these two definitions.