In this paper, the infinite sums of reciprocals and the partial sums derived from Chebyshev polynomials are studied. For the infinite sums of reciprocals, we apply the floor function to the reciprocals of these sums to obtain some new and interesting identities involving the Chebyshev polynomials. Simultaneously, we get several identities about the partial sums of Chebyshev polynomials by the relation of two types of Chebyshev polynomials.