Find the solution of the differential equation
\(\sin \theta \frac{\mathrm{d} y}{\mathrm{~d} \theta}+y=\tan \frac{1}{2} \theta,\)
where \(0\lt \theta\lt \pi\), given that \(y=1\) when \(\theta=\frac{1}{2} \pi\). Give your answer in the form \(y=\mathrm{f}(\theta)\).