Math Problem Statement

Verify csc²(θ) - cot²(θ) = cot(θ)tan(θ) is an identity.

Solution

To verify the given trigonometric identity:

csc2(θ)cot2(θ)=cot(θ)tan(θ)\csc^2(\theta) - \cot^2(\theta) = \cot(\theta)\tan(\theta)

Step-by-step solution:

1. Use a fundamental trigonometric identity:

csc2(θ)cot2(θ)=1\csc^2(\theta) - \cot^2(\theta) = 1 This is a standard trigonometric identity.

2. Substitute the identity into the left-hand side:

The left-hand side becomes: csc2(θ)cot2(θ)=1\csc^2(\theta) - \cot^2(\theta) = 1

3. Analyze the right-hand side:

The right-hand side is cot(θ)tan(θ)\cot(\theta)\tan(\theta). Recall the definitions of tan(θ)\tan(\theta) and cot(θ)\cot(\theta): tan(θ)=sin(θ)cos(θ),cot(θ)=cos(θ)sin(θ).\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}, \quad \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}. Thus: cot(θ)tan(θ)=(cos(θ)sin(θ))(sin(θ)cos(θ)).\cot(\theta)\tan(\theta) = \left(\frac{\cos(\theta)}{\sin(\theta)}\right)\left(\frac{\sin(\theta)}{\cos(\theta)}\right).

4. Simplify the right-hand side:

The terms sin(θ)\sin(\theta) and cos(θ)\cos(\theta) cancel out: cot(θ)tan(θ)=1.\cot(\theta)\tan(\theta) = 1.

5. Compare both sides:

The left-hand side and the right-hand side are equal: csc2(θ)cot2(θ)=cot(θ)tan(θ).\csc^2(\theta) - \cot^2(\theta) = \cot(\theta)\tan(\theta).

Conclusion:

The given equation is a valid identity.


Would you like a more detailed explanation or help with another step? Here are 5 related questions to explore further:

  1. What are the fundamental trigonometric identities and how can they be derived?
  2. How do we prove that csc2(θ)cot2(θ)=1\csc^2(\theta) - \cot^2(\theta) = 1?
  3. What are the applications of verifying trigonometric identities?
  4. How can the definitions of tan(θ)\tan(\theta) and cot(θ)\cot(\theta) be related to unit circle properties?
  5. What are common mistakes made when simplifying trigonometric expressions?

Tip:

When verifying trigonometric identities, always start by recalling key fundamental identities, such as the Pythagorean identities, to simplify your work.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Trigonometric Identities

Formulas

csc²(θ) - cot²(θ) = 1
tan(θ) = sin(θ)/cos(θ)
cot(θ) = cos(θ)/sin(θ)

Theorems

Pythagorean Identities

Suitable Grade Level

Grades 10-12