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You are looking at the sum of $m$-th powers of the imaginary parts of $n$-th roots of $-1.$ First, note that these can be expressed as polynomials in elementary symmetric functions (of the imaginary parts), so you just need to find the polynomial of which they are roots. Writing $$(x+i y)^n +1 = 0,$$ you get two equations, in $x$ and $y,$ eliminate $x$ and you will get the polynomial you want.