9758/2022/P1/Q04

4 (a) Show that $\frac{\mathrm{d}}{\mathrm{d} x}(\cot x)=-\operatorname{cosec}^2 x$.

[2]

(b) Show that $\sin 2 x \tan x=2 \sin ^2 x$.

[2]

(c) Hence, find the exact value of $\int_{\frac{\pi}{18}}^{\frac{\pi}{9}} \operatorname{cosec} 6 x \cot 3 x \mathrm{~d} x$.

[4]