Step 1: Write the given expression

We are given the expression for $y$: $y = \frac{\cos^2 A}{1 - \sin A}$

$y = \frac{\cos^2 A}{1 - \sin A}$

Step 2: Apply trigonometric identity for numerator

Using the fundamental trigonometric identity $\cos^2 A = 1 - \sin^2 A$, substitute it into the numerator: $y = \frac{1 - \sin^2 A}{1 - \sin A}$

$y = \frac{1 - \sin^2 A}{1 - \sin A}$

Step 3: Factorize the numerator

The numerator is in the form $a^2 - b^2 = (a - b)(a + b)$ where $a = 1$ and $b = \sin A$: $y = \frac{(1 - \sin A)(1 + \sin A)}{1 - \sin A}$

$y = \frac{(1 - \sin A)(1 + \sin A)}{1 - \sin A}$

Step 4: Cancel common terms and simplify

Since $\sin A \neq 1$, $1 - \sin A \neq 0$, so we can cancel $1 - \sin A$ from both numerator and denominator: $y = 1 + \sin A$

$y = 1 + \sin A$