https://file.aiquickdraw.com/mathbot/file/1727223559124tcnmjql6.jpg

The given expression is:

\[
\cos(x) \sin^2(x) + \cos^3(x)
\]

### Step-by-Step Simplification:

1. **Factor out** \(\cos(x)\) from both terms:
   \[
   \cos(x) \sin^2(x) + \cos^3(x) = \cos(x) \left( \sin^2(x) + \cos^2(x) \right)
   \]

2. **Use the Pythagorean identity**:
   \[
   \sin^2(x) + \cos^2(x) = 1
   \]
   Substituting this identity into the expression:
   \[
   \cos(x) \left( 1 \right) = \cos(x)
   \]

### Final Simplified Expression:
\[
\cos(x)
\]

The simplified form is \(\cos(x)\).

Would you like more details or have any questions? 

Here are 5 related questions for further practice:
1. Simplify \(\sin(x) \cos(x) + \cos^2(x)\).
2. Use trigonometric identities to simplify \(\sin^4(x) + \cos^4(x)\).
3. What is the simplified form of \(\sin(x) \cos^2(x) + \cos^3(x)\)?
4. How do you use the Pythagorean identity to simplify \(\sin^2(x) + 1 - \cos^2(x)\)?
5. Can you simplify \(\sec(x) \cos(x)\)?

### Tip:
When simplifying trigonometric expressions, always look for opportunities to apply the Pythagorean identity, \(\sin^2(x) + \cos^2(x) = 1\), to reduce the complexity of the expression.

Simplify cos(x) sin^2(x) + cos^3(x) using algebra and the fundamental trigonometric identities.

This guide provides a detailed solution on how to simplify the expression cos(x) sin^2(x) + cos^3(x) using the Pythagorean identity. Suitable for high school students learning advanced algebra and trigonometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation