(csc(alpha)+cot(alpha))/tan(alpha)+sin(alpha)= cot(alpha)csc(alpha)
Identify the erroneous step(s) (those that are NOT equal to the LHS)

To identify the erroneous step(s) in the given expression \(\frac{\csc(\alpha) + \cot(\alpha)}{\tan(\alpha)} + \sin(\alpha) = \cot(\alpha)\csc(\alpha)\), let's break down the problem step by step.

### Given Expression
\[
\frac{\csc(\alpha) + \cot(\alpha)}{\tan(\alpha)} + \sin(\alpha) = \cot(\alpha)\csc(\alpha)
\]

### Step-by-Step Simplification

1. **Rewrite the given trigonometric identities in terms of \(\sin(\alpha)\) and \(\cos(\alpha)\):**

   - \(\csc(\alpha) = \frac{1}{\sin(\alpha)}\)
   - \(\cot(\alpha) = \frac{\cos(\alpha)}{\sin(\alpha)}\)
   - \(\tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)}\)

2. **Substitute these identities into the given expression:**
\[
\frac{\frac{1}{\sin(\alpha)} + \frac{\cos(\alpha)}{\sin(\alpha)}}{\frac{\sin(\alpha)}{\cos(\alpha)}} + \sin(\alpha)
\]

3. **Combine the terms in the numerator:**
\[
\frac{\frac{1 + \cos(\alpha)}{\sin(\alpha)}}{\frac{\sin(\alpha)}{\cos(\alpha)}} + \sin(\alpha)
\]

4. **Simplify the complex fraction:**
\[
\frac{1 + \cos(\alpha)}{\sin(\alpha)} \cdot \frac{\cos(\alpha)}{\sin(\alpha)} + \sin(\alpha) = \frac{(1 + \cos(\alpha)) \cos(\alpha)}{\sin^2(\alpha)} + \sin(\alpha)
\]

5. **Combine the terms:**
\[
\frac{(1 + \cos(\alpha)) \cos(\alpha)}{\sin^2(\alpha)} + \sin(\alpha)
\]

### Compare with the Right-Hand Side
\[
\cot(\alpha) \csc(\alpha) = \frac{\cos(\alpha)}{\sin(\alpha)} \cdot \frac{1}{\sin(\alpha)} = \frac{\cos(\alpha)}{\sin^2(\alpha)}
\]

### Now, analyze:
\[
\frac{(1 + \cos(\alpha)) \cos(\alpha)}{\sin^2(\alpha)} + \sin(\alpha) \quad \text{vs.} \quad \frac{\cos(\alpha)}{\sin^2(\alpha)}
\]

Clearly, these two expressions are not equal, which means the left-hand side cannot be simplified to the right-hand side. Therefore, one of the steps (or the initial equality assumption) must be erroneous.

### Error Identification:
The erroneous step is assuming that:
\[
\frac{\csc(\alpha) + \cot(\alpha)}{\tan(\alpha)} + \sin(\alpha) = \cot(\alpha)\csc(\alpha)
\]
is true, since:
\[
\frac{(1 + \cos(\alpha)) \cos(\alpha)}{\sin^2(\alpha)} + \sin(\alpha) \neq \frac{\cos(\alpha)}{\sin^2(\alpha)}
\]

Thus, the error lies in the initial assumption itself. The step-by-step simplification shows that the left-hand side and the right-hand side do not match.

Would you like further details on any part of this solution or any other questions? Here are some related questions you might find useful:

1. How do you simplify trigonometric expressions?
2. What are the common trigonometric identities?
3. How do you convert between different trigonometric functions?
4. What is the significance of simplifying expressions in trigonometry?
5. How can you verify trigonometric identities?
6. What is the difference between \(\csc\), \(\cot\), and \(\tan\)?
7. How do you handle complex fractions in trigonometry?
8. What strategies can you use to solve trigonometric equations?

**Tip:** Always double-check each step when verifying trigonometric identities to ensure accuracy.

Explore how to identify erroneous steps in simplifying trigonometric expressions. This example focuses on a specific trigonometric identity involving sine, cosine, cosecant, and cotangent. Suitable for high school students and anyone interested in 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