$\begin{align}  & \text{From the left hand side} \\ & \frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta }{\sin \theta }=\frac{{{\sin }^{2}}\theta +{{(1+\cos \theta )}^{2}}}{\sin \theta (1+\cos \theta )} \\ & \frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta }{\sin \theta }=\frac{{{\sin }^{2}}\theta +{{\cos }^{2}}\theta +2\cos \theta +1}{\sin \theta (\cos \theta +1)} \\ & \frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta }{\sin \theta }=\frac{1+2\cos \theta +1}{\sin \theta (\cos \theta +1)} \\ & \frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta }{\sin \theta }=\frac{2\cos \theta +2}{\sin \theta (\cos \theta +1)} \\ & \frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta }{\sin \theta }=\frac{2(\cos \theta +1)}{\sin \theta (\cos \theta +1)}=\frac{2}{\sin \theta } \\ & \frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta }{\sin \theta }=2\operatorname{cosec}\theta  \\\end{align}$