Prove that : \[\frac{\sin \theta}{1+\cos \theta}+\frac{1+\cos \theta}{\sin \theta}=2 \operatorname{cosec} \theta \]

\[\begin{aligned} \mathrm{LHS} & =\frac{\sin ^{2} \theta+1+\cos ^{2} \theta+2 \cos \theta}{\sin \theta(1+\cos \theta)} \\ & =\frac{2+2 \cos \theta}{\sin \theta(1+\cos \theta)} \\ & =\frac{2}{\sin \theta} \\ & =2 \operatorname{cosec} \theta=\mathrm{RHS} \end{aligned} \]