2024 CCHY P1 Q13

Show that \(\sin 2x \cot x = 2 \cos^2 x\).[1]

Hence, without using a calculator, find the value of the constant \(k\) for which
\(\int_{\frac{\pi}{18}}^{\frac{\pi}{9}} \operatorname{cosec} 6x \tan 3x \; \mathrm{d}x = \frac{\sqrt{3}}{k}\).[5]