To verify the identity

\[\frac{\sin S}{1 + \cos A} + \frac{1 + \cos A}{\sin A} = 2 \csc A\]

I would start by writing $\csc A$ as $\large \frac{1}{\sin A}.$

Next multiply the numerator and denominator of the first fraction on the left side by $ 1 - \cos A$ and simplify. This will result in a denominator of $\sin^2 A.$ If you then multiply the numerator and denominator of the second fraction on the left side by $\sin A$ the two fractions will have the same denominator. Add the fractions and simplify.

Penny