Evaluate the following integrals: \[ \int_{\pi / 6}^{\pi / 2} \frac{\operatorname{cosec} x \cot ...

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Evaluate the following integrals:
\[
\int_{\pi / 6}^{\pi / 2} \frac{\operatorname{cosec} x \cot x}{1+\operatorname{cosec}^{2} x} d x
\]
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