We prove the more general identity

sinpz·sinp(z+1/n)·¼·sinp(z+(n-1)/n) = 21-nsinpnz.

(To derive the original identity, divide by sinpz and let z® 0.) Representing all sines as canonical products, we see that the left hand side is proportional to the right hand side. To find the coefficient of proportionality, we consider the asymptotic behavior when z = iy, y®+¥. We have sinp(iy+const) ~ exp(py)/2, from which the result follows.

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