Some Diophantine relations involving circular functions of rational angles

The eigenvalues of the 3 off-diagonal matrices of rank n with elements 1 + i cot [(j − k)π/n], sin[(j − k)π/n] and sin[(j − k)π/n], (j = 1, 2, . . . , n, k = 1, 2, . . . , n, j 6= k) are computed. The sums over k from 1 to n− 1 of cot (kπ/n) sin (2skπ/n) and sin(kπ/n)cos(2skπ/n) are moreover computed for s integer and p=2 and 4. The results are given by simple formulae in terms of integers. In this note we report some remarkable relations involving circular functions of rational angles. Theorem 1. The off-diagonal hermitian matrix of rank n whose elements are defined by the formula Ajk = (1− δjk) { 1 + i cot (j − k)π n }