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Question
\[\int\limits_{\pi/6}^{\pi/2} \frac{\ cosec x \cot x}{1 + {cosec}^2 x} dx\]
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Solution
\[\int_\frac{\pi}{6}^\frac{\pi}{2} \frac{\ cosecx\ cotx}{1 + \ cosec^2 x} d x\]
\[ = \int_\frac{\pi}{6}^\frac{\pi}{2} \frac{\ cosx}{1 + \sin^2 x} d x\]
\[ = \left[ \tan^{- 1} \left(\ sinx \right) \right]_\frac{\pi}{6}^\frac{\pi}{2} \]
\[ = \tan^{- 1} 1 - \tan^{- 1} \frac{1}{2}\]
\[ = \tan^{- 1} \frac{1 - \frac{1}{2}}{1 + 1 \times \frac{1}{2}}\]
\[ = \tan^{- 1} \frac{1}{3}\]
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