https://file.aiquickdraw.com/mathbot/file/1728828823898an92hwee.jpg

The given expression is:

\[
\frac{\cos^2{x} + 16 \cos{x} + 64}{\cos{x} + 8}
\]

### Step-by-Step Simplification:

1. **Factor the numerator**:
   The numerator is a quadratic expression in terms of \(\cos{x}\):
   \[
   \cos^2{x} + 16\cos{x} + 64
   \]
   This can be factored as:
   \[
   (\cos{x} + 8)(\cos{x} + 8)
   \]
   or:
   \[
   (\cos{x} + 8)^2
   \]

2. **Rewrite the expression**:
   Now substitute this factorization into the original expression:
   \[
   \frac{(\cos{x} + 8)^2}{\cos{x} + 8}
   \]

3. **Simplify the fraction**:
   Since \(\cos{x} + 8\) appears in both the numerator and the denominator, we can cancel one factor:
   \[
   \cos{x} + 8
   \]

Thus, the simplified expression is:

\[
\cos{x} + 8
\]

Would you like further clarification or details?

### Related Questions:
1. How do you factor quadratic expressions in trigonometry?
2. Can \(\cos{x} + 8 = 0\) lead to any restrictions in this problem?
3. How does simplifying trigonometric expressions help in solving equations?
4. What are the steps to factor quadratic expressions generally?
5. How do cancellations work in simplifying fractions?

### Tip:
Always check if the expression you cancel from both the numerator and denominator can be zero to avoid undefined expressions.

Simplify the expression: (cos^2(x) + 16 cos(x) + 64) / (cos(x) + 8).

Learn how to simplify the trigonometric expression (cos^2(x) + 16 cos(x) + 64) / (cos(x) + 8) by factoring the quadratic numerator and canceling common terms. This problem covers key concepts in trigonometry and algebra, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation