Answer
$$\frac{1}{2}$$
Work Step by Step
$$\eqalign{
& \int_{\pi /12}^{\pi /4} {\csc 2x\cot 2x} dx \cr
& {\text{Let }}u = 2x,{\text{ }}du = 2dx \cr
& {\text{The new limits of integration are:}} \cr
& x = \frac{\pi }{4} \to u = 2\left( {\frac{\pi }{4}} \right) = \frac{\pi }{2} \cr
& x = \frac{\pi }{{12}} \to u = 2\left( {\frac{\pi }{{12}}} \right) = \frac{\pi }{6} \cr
& {\text{Substituting}} \cr
& \int_{\pi /12}^{\pi /4} {\csc 2x\cot 2x} dx = \int_{\pi /6}^{\pi /2} {\csc u\cot u\left( {\frac{1}{2}} \right)} du \cr
& = \frac{1}{2}\int_{\pi /6}^{\pi /2} {\csc u\cot u} du \cr
& {\text{Integrating}} \cr
& = - \frac{1}{2}\left[ {\csc u} \right]_{\pi /6}^{\pi /2} \cr
& = - \frac{1}{2}\left[ {\csc \left( {\frac{\pi }{2}} \right) - \csc \left( {\frac{\pi }{6}} \right)} \right] \cr
& = - \frac{1}{2}\left( {1 - 2} \right) \cr
& = \frac{1}{2} \cr} $$
