Show that \(2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) \equiv \sec ^ { 2 } x\).
Solve the equation \(2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) = \tan x + 21\) for \(0 < x < \pi\), giving your answers correct to 3 significant figures.
Find \(\int \left[ 2 \operatorname { cosec } ^ { 2 } ( 4 y + 2 ) - 2 \operatorname { cosec } ^ { 2 } ( 4 y + 2 ) \cos ( 4 y + 2 ) \right] \mathrm { d } y\).
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