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Let $n$ be any positive odd integer. We show that $$\\sum_{r=0}^{n-1}\\frac1{1+\\sin2\\pi\\frac{x+r}n+\\cos2\\pi\\frac{x+r}n} =\\frac{(-1)^{(n-1)/2}n}{1+(-1)^{(n-1)/2}\\sin 2\\pi x+\\cos 2\\pi x}$$ for any complex number with $x+1/2,x+(-1)^{(n-1)/2}/4\\not\\in\\mathbb Z$, and $$\\sum_{j,k=0}^{n-1}\\frac1{\\sin 2\\pi\\frac{x+j}n+\\sin2\\pi \\frac{y+k}n}=\\frac{(-1)^{(n-1)/2}n^2}{\\sin 2\\pi x+\\sin2\\pi y}$$ for all complex numbers $x$ and $y$ with $x+y,x-y-1/2\\not\\in\\mathbb Z$. 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