91Ó°ÊÓ

The concept of the difference of squares is an essential algebraic tool that simplifies expressions.
This happens when you have two squared terms separated by a subtraction sign. The general form is:
\[ a^2 - b^2 = (a + b)(a - b) \]
In our original exercise, the expression \[ \text{sin}^2 x - \text{cos}^2 x \] is a perfect example. We can rewrite it as:
\[ (\text{sin} x + \text{cos} x)(\text{sin} x - \text{cos} x) \]
By decomposing it this way, we make it easier to simplify further.

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equation are defined. These identities are like tools in a toolbox for simplifying trigonometric expressions and solving equations.
One such identity used in the exercise is the Pythagorean identity:
\[ \text{sin}^2 x + \text{cos}^2 x = 1 \]
While this specific identity wasn’t used directly in the exercise solution, recognizing these identities helps in understanding the relationships between different trigonometric functions.
The manipulation of \[ \text {sin}^{2} x - \text {cos}^{2} x \] relies on knowing these foundational identities and how they interplay with algebraic techniques like factoring.

Simplification of fractions involves reducing fractions to their simplest form. This simplifies not only numerical fractions but also algebraic fractions involving variables and functions.
In our exercise, after factoring, we have:
\[ \frac{(\text{sin} x + \text{cos} x)(\text{sin} x - \text{cos} x)}{\text{sin} x - \text{cos} x} \]
Because \text{sin} x - \text{cos} x \ is both in the numerator and the denominator, it can be cancelled out:
\[ \frac{(\text{sin} x - \text{cos} x)}{(\text{sin} x - \text{cos} x)} = 1 \]
Leaving us with: \[ \text{sin} x + \text{cos} x \]
This process of cancelling common factors makes the expression more manageable and straightforward.