Identities for combinatorial sums involving trigonometric functions

LetA(m,n)(a) = sigma(m) (j=0) (-4)(j )(m + j 2j) sigma(n-1) (k=0)sin(a + 2k pi/n) cos(2j)(a + 2k pi/n)andB(m,n)(a) = sigma(m) (j=0) (-4)(j)(m + j +1 2j+1)sigma(n-1 ) (k=0 )sin(a + 2k pi/n)x cos(2j+1)(a + 2k pi/n),where m >= 0 and n >= 1 are integers and a is a real number. We present two proofs for the following results:(i) If 2m + 1 equivalent to 0 (mod n), thenA(m,n)(a) = (-1)(m)n sin((2m + 1)a).(ii) If 2m + 1 (sic) 0 (mod n), then A(m,n)(a) = 0.(iii) If 2(m + 1) equivalent to 0 (mod n), thenB(m,n)(a) = (-1)(m) n/2 sin(2(m + 1)a).(iv) If 2(m + 1) (sic) 0 (mod n), then B-m,B-n(a) = 0.