Math Problem Statement

(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)

Solution

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

Given Expression

csc(α)+cot(α)tan(α)+sin(α)=cot(α)csc(α)\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(α)\sin(\alpha) and cos(α)\cos(\alpha):

    • csc(α)=1sin(α)\csc(\alpha) = \frac{1}{\sin(\alpha)}
    • cot(α)=cos(α)sin(α)\cot(\alpha) = \frac{\cos(\alpha)}{\sin(\alpha)}
    • tan(α)=sin(α)cos(α)\tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)}
  2. Substitute these identities into the given expression: 1sin(α)+cos(α)sin(α)sin(α)cos(α)+sin(α)\frac{\frac{1}{\sin(\alpha)} + \frac{\cos(\alpha)}{\sin(\alpha)}}{\frac{\sin(\alpha)}{\cos(\alpha)}} + \sin(\alpha)

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

  4. Simplify the complex fraction: 1+cos(α)sin(α)cos(α)sin(α)+sin(α)=(1+cos(α))cos(α)sin2(α)+sin(α)\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: (1+cos(α))cos(α)sin2(α)+sin(α)\frac{(1 + \cos(\alpha)) \cos(\alpha)}{\sin^2(\alpha)} + \sin(\alpha)

Compare with the Right-Hand Side

cot(α)csc(α)=cos(α)sin(α)1sin(α)=cos(α)sin2(α)\cot(\alpha) \csc(\alpha) = \frac{\cos(\alpha)}{\sin(\alpha)} \cdot \frac{1}{\sin(\alpha)} = \frac{\cos(\alpha)}{\sin^2(\alpha)}

Now, analyze:

(1+cos(α))cos(α)sin2(α)+sin(α)vs.cos(α)sin2(α)\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: csc(α)+cot(α)tan(α)+sin(α)=cot(α)csc(α)\frac{\csc(\alpha) + \cot(\alpha)}{\tan(\alpha)} + \sin(\alpha) = \cot(\alpha)\csc(\alpha) is true, since: (1+cos(α))cos(α)sin2(α)+sin(α)cos(α)sin2(α)\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\csc, cot\cot, and tan\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.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Trigonometric Identities
Complex Fractions

Formulas

Trigonometric identities involving sine, cosine, cosecant, cotangent, tangent

Theorems

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Suitable Grade Level

High School