(i) Prove the identity \(\cot x - \cot 2x \equiv \csc 2x\).

(ii) Show that \(\int_{\frac{1}{6}\pi}^{\frac{1}{4}\pi} \cot x \, dx = \frac{1}{2} \ln 2\).

(iii) Find the exact value of \(\int_{\frac{1}{6}\pi}^{\frac{1}{4}\pi} \csc 2x \, dx\), giving your answer in the form \(a \ln b\).