7 The variables \(x\) and \(\theta\) satisfy the differential equation
$$x \sin ^ { 2 } \theta \frac { \mathrm {~d} x } { \mathrm {~d} \theta } = \tan ^ { 2 } \theta - 2 \cot \theta$$
for \(0 < \theta < \frac { 1 } { 2 } \pi\) and \(x > 0\). It is given that \(x = 2\) when \(\theta = \frac { 1 } { 4 } \pi\).
- Show that \(\frac { \mathrm { d } } { \mathrm { d } \theta } \left( \cot ^ { 2 } \theta \right) = - \frac { 2 \cot \theta } { \sin ^ { 2 } \theta }\).
(You may assume without proof that the derivative of \(\cot \theta\) with respect to \(\theta\) is \(- \operatorname { cosec } ^ { 2 } \theta\).) - Solve the differential equation and find the value of \(x\) when \(\theta = \frac { 1 } { 6 } \pi\).