verify the identity by starting with the more complicated side of the identity and transforming it into the simpler side: 1+cos(theta) = (sin^2(theta))/(1-cos(theta))

To verify the identity, we start with the more complicated side, \( \frac{\sin^2(\theta)}{1 - \cos(\theta)} \), and transform it into \( 1 + \cos(\theta) \).

### Step-by-Step Solution:
1. **Rewrite \( \sin^2(\theta) \):**  
   Using the Pythagorean identity, \( \sin^2(\theta) = 1 - \cos^2(\theta) \):
   \[
   \frac{\sin^2(\theta)}{1 - \cos(\theta)} = \frac{1 - \cos^2(\theta)}{1 - \cos(\theta)}
   \]

2. **Factor the Numerator:**  
   The numerator \( 1 - \cos^2(\theta) \) is a difference of squares, which can be factored as:
   \[
   1 - \cos^2(\theta) = (1 - \cos(\theta))(1 + \cos(\theta))
   \]
   So we substitute this into the expression:
   \[
   \frac{(1 - \cos(\theta))(1 + \cos(\theta))}{1 - \cos(\theta)}
   \]

3. **Cancel Out \( 1 - \cos(\theta) \):**  
   Since \( 1 - \cos(\theta) \) appears in both the numerator and the denominator, we can cancel it (assuming \( \cos(\theta) \neq 1 \)):
   \[
   = 1 + \cos(\theta)
   \]

### Conclusion:
After simplifying, we find:
\[
\frac{\sin^2(\theta)}{1 - \cos(\theta)} = 1 + \cos(\theta)
\]
Thus, the identity is verified.

---

Would you like more details on any step or have questions?

### Related Questions
1. What are other common trigonometric identities involving \( \sin(\theta) \) and \( \cos(\theta) \)?
2. How does the difference of squares factorization help in simplifying trigonometric expressions?
3. What are common steps to verify trigonometric identities?
4. How can Pythagorean identities be used in complex simplifications?
5. When do we need to be cautious about dividing by trigonometric expressions?

### Tip
When verifying trigonometric identities, starting with the more complicated side usually reveals simplification strategies like factoring or applying known identities.

Verify the identity by starting with the more complicated side of the identity and transforming it into the simpler side: 1 + cos(theta) = (sin^2(theta))/(1 - cos(theta))

Explore how to verify the trigonometric identity 1 + cos(theta) = (sin^2(theta))/(1 - cos(theta)) through a detailed step-by-step solution. This problem covers essential trigonometric concepts, including the Pythagorean identity and difference of squares, aimed at students in Grades 10-12.

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