91Ó°ÊÓ

Factoring is a key step in simplifying trigonometric expressions. In our given exercise, we start with the expression \( \frac{\text{sin} x - \text{sin}^2 x}{\text{sin} x} \). When we look at the numerator \(\text{sin} x - \text{sin}^2 x\), we can see that it has a common factor of \(\text{sin} x\) in both terms.

Think about factoring similar to pulling out the greatest common divisor in algebra. Here’s how it works:

• If you have two terms, like \(\text{sin} x\) and \(\text{sin}^2 x\), we notice that \(\text{sin} x\) is present in both.
• Therefore, we can factor out \(\text{sin} x\), rewriting \(\text{sin} x - \text{sin}^2 x\) as \(\text{sin} x (1 - \text{sin} x)\).

This simpler form reveals the structure of the expression and prepares it for the next steps in simplifying.

After factoring, the next goal is to simplify the fraction. Rewriting the initial expression \( \frac{\text{sin} x - \text{sin}^2 x}{\text{sin} x} \) using the factored terms, we get \( \frac{\text{sin} x (1 - \text{sin} x)}{\text{sin} x} \).

This step shows the numerator and denominator more clearly. Simplifying trigonometric fractions follows the same principles as simplifying any type of fraction. By factoring the numerator and rewriting the expression, we break down complex fractions into easier-to-handle pieces.

• Rewrite the numerators or denominators in terms of products if possible.
• Simplification often makes invisible common factors visible, by writing them as products.

This sets up a perfect stage to 'cancel' factors that appear in both the numerator and denominator.

The final stage in simplifying our expression involves canceling common factors. From our factored form \( \frac{\text{sin} x (1 - \text{sin} x)}{\text{sin} x} \), we see \(\text{sin} x\) appears in both the numerator and the denominator. Just like when you cancel common numbers in arithmetic fractions, you can cancel common factors in trigonometric fractions too.

By doing this, we remove \(\text{sin} x\) from both the top and the bottom, resulting in \( 1 - \text{sin} x \).

• Remove common factors that appear in both top and bottom.
• Only factors (not terms separated by addition or subtraction) can be canceled.

Canceling helps reduce the fraction to its simplest form. The initial complicated fraction is now the much simpler result \(1 - \text{sin} x\). Breaking down expressions step-by-step in this way can make even complex simplifications approachable.