https://file.aiquickdraw.com/mathbot/file/1730767234002g6xoh6rz.jpg

To simplify the expression \((1 + \cos x)(1 - \cos x)\), let's proceed as follows:

### Step-by-Step Solution
The expression \((1 + \cos x)(1 - \cos x)\) can be identified as a difference of squares, since it has the form \((a + b)(a - b) = a^2 - b^2\).

1. Here, \(a = 1\) and \(b = \cos x\).
   
2. Using the difference of squares formula:
   \[
   (1 + \cos x)(1 - \cos x) = 1^2 - (\cos x)^2
   \]

3. Simplify further:
   \[
   1 - \cos^2 x
   \]

4. Now, we use the Pythagorean identity for trigonometric functions, which states:
   \[
   \sin^2 x + \cos^2 x = 1
   \]
   Rearranging this identity, we get:
   \[
   \sin^2 x = 1 - \cos^2 x
   \]

5. Substitute \(1 - \cos^2 x\) with \(\sin^2 x\):
   \[
   1 - \cos^2 x = \sin^2 x
   \]

### Final Answer
Thus, \((1 + \cos x)(1 - \cos x) = \sin^2 x\).

### Correct Choice
The correct answer is:
**c) \(\sin^2 x\)**

Would you like further details or have any questions?

---

Here are some related questions to expand on this concept:

1. How does the difference of squares formula apply in trigonometric simplifications?
2. Can the Pythagorean identity be rearranged to solve for \(\cos^2 x\) in terms of \(\sin^2 x\)?
3. What other trigonometric identities could simplify expressions involving both \(\sin\) and \(\cos\)?
4. How would you simplify \((1 - \sin x)(1 + \sin x)\)?
5. Could you apply a similar process to verify if \((\tan x + \cot x)(\tan x - \cot x)\) has a simplification?

---

**Tip:** Always look for familiar algebraic patterns, like the difference of squares, when simplifying trigonometric expressions.

Determine the correct simplification of the expression: (1 + cos x)(1 − cos x)

Learn how to simplify the trigonometric expression (1 + cos x)(1 - cos x) using algebraic and trigonometric identities. This solution explains the difference of squares formula and the Pythagorean identity, guiding students in Grades 10-12 through the simplification process.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation