Apply algebraic identity

Use the algebraic difference of squares formula $(x + y)(x - y) = x^2 - y^2$ on the first two terms $(1 + \sin A)(1 - \sin A)$.

$(1 + \sin A)(1 - \sin A) = 1 - \sin^2 A$

Apply fundamental trigonometric identities

Recall the standard Pythagorean trigonometric identities: $1 - \sin^2 A = \cos^2 A$ and $1 + \tan^2 A = \sec^2 A$. Substitute these into the expression.

$(1 - \sin^2 A)(1 + \tan^2 A) = \cos^2 A \cdot \sec^2 A$

Simplify using reciprocal identity

Since $\sec A = \frac{1}{\cos A}$, it follows that $\sec^2 A = \frac{1}{\cos^2 A}$.

$\cos^2 A \cdot \sec^2 A = \cos^2 A \cdot \frac{1}{\cos^2 A} = 1$