Identities for combinatorial sums involving trigonometric functions

Let $A_{m,n}(a)=\sum_{j=0}^m (-4)^j {m+j\choose 2j}\sum_{k=0}^{n-1} \sin(a+2kπ/n) \cos^{2j}(a+2kπ/n)$ and $B_{m,n}(a)=\sum_{j=0}^m (-4)^j {m+j+1\choose 2j+1}\sum_{k=0}^{n-1} \sin(a+2kπ/n) \cos^{2j+1}(a+2kπ/n),$ where $m\geq 0$ and $n\geq 1$ are integers and $a$ is a real number. We present two proofs for the following results: (i) If $2m+1 \equiv 0 \, (\mbox{mod} \, n)$, then $A_{m,n}(a)=(-1)^m n \sin((2m+1)a).$ (ii) If $2m+1 \not\equiv 0 \, (\mbox{mod} \, n)$, then $A_{m,n}(a)=0$. (iii) If $2(m+1) \equiv 0 \, (\mbox{mod} \, n)$, then $B_{m,n}(a)=(-1)^m \frac{n}{2} \sin(2(m+1)a).$ (iv) If $2(m+1) \not\equiv 0 \, (\mbox{mod} \, n)$, then $B_{m,n}(a)=0$.

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