First order differential equations (integrating factor)
5
Differentiate \(\frac { 1 } { \sin ^ { 2 } \theta }\) with respect to \(\theta\).
The variables \(x\) and \(\theta\) satisfy the differential equation
$$x \tan \theta \frac { d x } { d \theta } + \operatorname { cosec } ^ { 2 } \theta = 0$$
for \(0 < \theta < \frac { 1 } { 2 } \pi\) and \(x > 0\). It is given that \(x = 4\) when \(\theta = \frac { 1 } { 6 } \pi\). Solve the differential equation, obtaining an expression for \(x\) in terms of \(\theta\).