In this paper, in view of a determinantal formula for higher order derivatives of the ratio of two differentiable functions, we expand the logarithm of the normalized tail of the power series expansion of the cosine function into a Maclaurin power series expansion whose coefficients are expressed in terms of specific Hessenberg determinants, present the decreasing property and concavity of the normalized tail of the Maclaurin power series expansion of the cosine function, deduce a new determinantal expression of the Bernoulli numbers, and verify the decreasing property for the ratio of the logarithms of the first two normalized tails of the Maclaurin power series expansion of the cosine function.

In the paper, the authors simply review recent results of inequalities, monotonicity, signs of determinants, determinantal formulas, closed-form expressions, and identities of the Bernoulli numbers and polynomials, establish an identity involving the differences between the Bernoulli polynomials and the Bernoulli numbers, present two identities among the differences between the Bernoulli polynomials and the Bernoulli numbers in terms of a determinant and a partial Bell polynomial, and derive a determinantal formula of the differences between the Bernoulli polynomials and the Bernoulli numbers.