Two identities and closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind

In this article, the authors present two identities involving products of the Bernoulli numbers, provide four alternative proofs for these two identities, derive two closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind, and supply simple proofs of series expansions of (hyperbolic) cosecant and cotangent functions.