lim x close to 0 of (cosx-cos3x)/(1-cos^2x)

We are tasked with evaluating the limit:

\[
\lim_{x \to 0} \frac{\cos(x) - \cos(3x)}{1 - \cos^2(x)}.
\]

### Step-by-Step Solution

1. **Simplify the Denominator**:  
   Notice that \(1 - \cos^2(x)\) is a well-known trigonometric identity:
   \[
   1 - \cos^2(x) = \sin^2(x).
   \]
   So, the expression becomes:
   \[
   \frac{\cos(x) - \cos(3x)}{\sin^2(x)}.
   \]

2. **Expand the Numerator**:  
   We can use the sum-to-product formula for the difference of cosines:
   \[
   \cos(A) - \cos(B) = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right).
   \]
   Applying this to \(\cos(x) - \cos(3x)\), we get:
   \[
   \cos(x) - \cos(3x) = -2 \sin\left(\frac{x + 3x}{2}\right) \sin\left(\frac{x - 3x}{2}\right)
   = -2 \sin(2x) \sin(-x) = 2 \sin(2x) \sin(x).
   \]
   So, the expression becomes:
   \[
   \frac{2 \sin(2x) \sin(x)}{\sin^2(x)}.
   \]

3. **Simplify the Expression**:
   Cancel one \(\sin(x)\) from the numerator and denominator:
   \[
   \frac{2 \sin(2x)}{\sin(x)}.
   \]
   Using the identity \(\sin(2x) = 2 \sin(x) \cos(x)\), we substitute:
   \[
   \frac{2 \cdot 2 \sin(x) \cos(x)}{\sin(x)} = 4 \cos(x).
   \]

4. **Evaluate the Limit**:  
   Now, evaluate the limit as \(x \to 0\):
   \[
   \lim_{x \to 0} 4 \cos(x) = 4 \cos(0) = 4 \cdot 1 = 4.
   \]

### Final Answer:
\[
\lim_{x \to 0} \frac{\cos(x) - \cos(3x)}{1 - \cos^2(x)} = 4.
\]

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. What are the sum-to-product formulas in trigonometry, and how are they derived?
2. How would you compute the limit if the numerator was \(\sin(x) - \sin(3x)\) instead?
3. Why is \(\sin^2(x)\) used frequently in limit problems, especially around \(x = 0\)?
4. How would you approach limits that involve higher powers of \(\sin(x)\) or \(\cos(x)\)?
5. How can L'Hopital's Rule be applied to similar trigonometric limits?

**Tip**: When dealing with trigonometric limits, always check for identities like small angle approximations or product-to-sum formulas.

lim x close to 0 of (cosx - cos3x)/(1 - cos^2x)

Explore the step-by-step solution to the limit problem: lim x → 0 of (cos(x) - cos(3x)) / (1 - cos^2(x)). This problem involves key trigonometric identities, such as the sum-to-product formula and small angle approximations. Suitable for students in Grades 11-12 and college.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation