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Pythagorean identities are fundamental relationships in trigonometry that describe the intrinsic properties of sine and cosine functions. They originate from the Pythagorean Theorem, which relates the sides of a right triangle.

The most common Pythagorean identity is \[ \sin^2(y) + \cos^2(y) = 1 \.\] From this equation, we can derive other useful forms, such as \[ \sin^2(y) = 1 - \cos^2(y) \]\ and \[ \cos^2(y) = 1 - \sin^2(y) \.\] Having these identities in hand allows us to transform trigonometric expressions and resolve them into simpler forms.

When tackling the given exercise, recognition of the appropriate identity allows the re-writing of the original fraction for simplification purposes. Thus, students should commit these identities to memory and practice identifying opportunities to apply them in various problem scenarios.

Trigonometric simplification often involves combining multiple trigonometric terms into a single term or simpler expression using identities and algebraic manipulation.

For the exercise provided, the simplification process begins with the substitution of the Pythagorean identity into the given equation. This reduces the complexity of the problem, converting a fractional expression containing a trigonometric function into an equivalent expression with a polynomial form.

This step is crucial because it transforms our expression into something we can factor, a necessary precursor to cancellation. Factoring out common terms becomes the key to reducing our expression to its simplest form.

Factoring is an algebraic process used to break down polynomials into products of their factors. It is a critical tool when simplifying expressions, especially when working with fractions. In the context of the exercise, we use factoring to remove a common term from both the numerator and denominator of a fraction, which simplifies the overall expression.

As exemplified by Step 3 in the solution, the numerator \(1 - \cos^2(y)\) factors into \(1 - \cos(y)\) times \(1 + \cos(y)\). By observing that the denominator \(1 - \cos(y)\) is a factor of the numerator, we can then cancel this term from both parts of the fraction.

It's essential to understand that factoring is not just about finding two numbers that multiply to the original term but identifying factors that reveal structure and simplify the expression. In the context of simplifying trigonometric expressions, factoring coupled with the application of identities forms the basis for most simplifications.