Contour integration of a trigonometric function

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Following is true.(My book says..) $$2 - \frac12 h = \frac{h^2}{2\pi}\int^{\pi / h}_{-\pi / h}\sin ^2\xi \cot^2\frac12 h \xi \, d\xi $$

Using contour integration, how to show this is true? Can you show the process?