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.