Question

\[\int\limits_{\pi/6}^{\pi/2} \frac{\ cosec x \cot x}{1 + {cosec}^2 x} dx\]

Answer 1

\[\int_\frac{\pi}{6}^\frac{\pi}{2} \frac{\ cosecx\ cotx}{1 + \ cosec^2 x} d x\]

\[ = \int_\frac{\pi}{6}^\frac{\pi}{2} \frac{\ cosx}{1 + \sin^2 x} d x\]

\[ = \left[ \tan^{- 1} \left(\ sinx \right) \right]_\frac{\pi}{6}^\frac{\pi}{2} \]

\[ = \tan^{- 1} 1 - \tan^{- 1} \frac{1}{2}\]

\[ = \tan^{- 1} \frac{1 - \frac{1}{2}}{1 + 1 \times \frac{1}{2}}\]

\[ = \tan^{- 1} \frac{1}{3}\]