Remarkable identities

In the paper a number of identities involving even powers of the values of functions tangent, cotangent, secans and cosecans are proved. Namely, the following relations are shown: [wzory] where m, n are positive integers, f is one of the functions: tangent, cotangent, secans or cosecans and wf(x),vf(x),~wf(x) are some polynomials from Q[x]. One of the remarkable identities is the following: [wzór] Some of these identities are used to find, by elementary means, the sums of the series of the form [wzór] , where n is a fixed positive integer. One can also notice that Bernoulli numbers appear in the leading coeficients of the polynomials wf(x),vf(x) and ~ wf(x).