We show that the sequences of polynomials with zeros
cot(mπ/(n+2))
and tan(mπ/(n+2))
are not
orthogonal sequences with respect to any integral inner product. We give an algebraic formula for these polynomials, that is
simpler than the formula originally derived by
Cvijovic and Klinowski (1998). New sequences of
polynomials with algebraic numbers as roots and closed
trigonometric formulas are also derived by these methods.