Values of the derivatives of the cotangent at rational multiples of π

By elementary arguments, we deduce closed-form expressions for the values of all derivatives of the cotangent function at rational multiples of π . These formulae are considerably simpler than similar ones which were found in a different manner by Kölbig. Also, we show that the values of cot ( n ) ( π x ) , n ∈ N , at x = 1 2 , 1 3 , 2 3 , 1 4 , 3 4 , 1 6  and  5 6 are expressible in terms of the values of the Bernoulli polynomials alone.