Taking the common denominator

To combine the fractions, multiply the denominators to get $\sin A (1 + \cos A)$.

$\frac{\sin^2 A + (1 + \cos A)^2}{\sin A (1 + \cos A)}$

Expanding the numerator

Expand $(1 + \cos A)^2$ as $1 + 2 \cos A + \cos^2 A$.

$\frac{\sin^2 A + 1 + 2 \cos A + \cos^2 A}{\sin A (1 + \cos A)}$

Applying trigonometric identity

Replace $\sin^2 A + \cos^2 A$ with $1$. The expression becomes $\frac{1 + 1 + 2 \cos A}{\sin A (1 + \cos A)}$.

$\frac{2 + 2 \cos A}{\sin A (1 + \cos A)} = \frac{2(1 + \cos A)}{\sin A (1 + \cos A)}$

Simplifying and final result

Cancel the common term $(1 + \cos A)$ from numerator and denominator to get $\frac{2}{\sin A}$, which is $2 \csc A$.

$\frac{2}{\sin A} = 2 \csc A$