Math Problem Statement

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Prove the identities: \( \frac{\cot \theta}{\csc \theta} = \cos \theta \) and \( \csc \theta - \sin \theta = \cos \theta \cot \theta \).

Solution

Let's tackle each problem step-by-step, showing all work and reasoning to help you follow along.

Problem 14

Prove the identity: cot⁡θcsc⁡θ=cos⁡θ\frac{\cot \theta}{\csc \theta} = \cos \theta

  1. Rewrite cot⁡θ\cot \theta and csc⁡θ\csc \theta in terms of sine and cosine: cot⁡θ=cos⁡θsin⁡θandcsc⁡θ=1sin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta} \quad \text{and} \quad \csc \theta = \frac{1}{\sin \theta}

  2. Substitute these values into the left side of the identity: cot⁡θcsc⁡θ=cos⁡θsin⁡θ1sin⁡θ\frac{\cot \theta}{\csc \theta} = \frac{\frac{\cos \theta}{\sin \theta}}{\frac{1}{\sin \theta}}

  3. Simplify the fraction by multiplying the numerator by the reciprocal of the denominator: =cos⁡θsin⁡θ⋅sin⁡θ1= \frac{\cos \theta}{\sin \theta} \cdot \frac{\sin \theta}{1}

  4. Cancel sin⁡θ\sin \theta from the numerator and denominator: =cos⁡θ= \cos \theta

  5. Conclusion: cot⁡θcsc⁡θ=cos⁡θ\frac{\cot \theta}{\csc \theta} = \cos \theta This proves the identity.

Problem 15

Prove the identity: csc⁡θ−sin⁡θ=cos⁡θcot⁡θ\csc \theta - \sin \theta = \cos \theta \cot \theta

  1. Rewrite csc⁡θ\csc \theta, sin⁡θ\sin \theta, and cot⁡θ\cot \theta in terms of sine and cosine: csc⁡θ=1sin⁡θ,sin⁡θ=sin⁡θ,andcot⁡θ=cos⁡θsin⁡θ\csc \theta = \frac{1}{\sin \theta}, \quad \sin \theta = \sin \theta, \quad \text{and} \quad \cot \theta = \frac{\cos \theta}{\sin \theta}

  2. Substitute these values into the left side of the identity: csc⁡θ−sin⁡θ=1sin⁡θ−sin⁡θ\csc \theta - \sin \theta = \frac{1}{\sin \theta} - \sin \theta

  3. Rewrite sin⁡θ\sin \theta as a fraction with a common denominator: =1sin⁡θ−sin⁡2θsin⁡θ= \frac{1}{\sin \theta} - \frac{\sin^2 \theta}{\sin \theta}

  4. Combine the terms over a common denominator: =1−sin⁡2θsin⁡θ= \frac{1 - \sin^2 \theta}{\sin \theta}

  5. Use the Pythagorean identity 1−sin⁡2θ=cos⁡2θ1 - \sin^2 \theta = \cos^2 \theta: =cos⁡2θsin⁡θ= \frac{\cos^2 \theta}{\sin \theta}

  6. Separate the fraction as cos⁡θ⋅cos⁡θsin⁡θ\cos \theta \cdot \frac{\cos \theta}{\sin \theta}: =cos⁡θ⋅cot⁡θ= \cos \theta \cdot \cot \theta

  7. Conclusion: csc⁡θ−sin⁡θ=cos⁡θcot⁡θ\csc \theta - \sin \theta = \cos \theta \cot \theta This proves the identity.

Would you like more explanation on any step, or do you have any questions?

Related Questions

  1. How can I use Pythagorean identities to simplify trigonometric expressions?
  2. What are common strategies for proving trigonometric identities?
  3. How do reciprocal identities work with cotangent, secant, and cosecant?
  4. What are some other examples of identities involving cot⁡\cot and csc⁡\csc?
  5. How does one determine which identity to use when simplifying trigonometric expressions?

Tip

When proving trigonometric identities, try to convert everything into sine and cosine. This approach often reveals simplifications that may not be immediately obvious in the original form.

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Math Problem Analysis

Mathematical Concepts

Trigonometric Identities
Simplification
Reciprocal and Quotient Identities

Formulas

\( \cot \theta = \frac{\cos \theta}{\sin \theta} \)
\( \csc \theta = \frac{1}{\sin \theta} \)
Pythagorean Identity: \( 1 - \sin^2 \theta = \cos^2 \theta \)

Theorems

Trigonometric Identity

Suitable Grade Level

Grades 10-12

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