In this paper, the Bezout's identity is analyzed in the context of classical orthogonal polynomials solution of a second order differential equation of hypergeometric type. Differential equations, relation with the starting family as well as recurrence relations and explicit representations are given for the Bezout's pair. Extensions to classical orthogonal polynomials of a discrete variable and their q-analogues are also presented. Applications of these results for the representation of the second kind functions are given.