Trigonometric Sums via Lagrange Interpolation

By means of the Lagrange interpolation, we derive two trigonometric identities that are utilized to evaluate, in closed forms, eight classes of power sums of trigonometric functions over equally distributed angles around the unit circle. When the mth term is removed from the sums, four classes (among eight) are shown to admit also closed expressions, that are surprisingly independent of the integer parameter “m”.