Show that \(\sin 2 x \cot x \equiv 2 \cos ^ { 2 } x\).
Using the identity in part (i),
(a) find the least possible value of
$$3 \sin 2 x \cot x + 5 \cos 2 x + 8$$
as \(x\) varies,
(b) find the exact value of \(\int _ { \frac { 1 } { 8 } \pi } ^ { \frac { 1 } { 6 } \pi } \operatorname { cosec } 4 x \tan 2 x \mathrm {~d} x\).