Use properties of natural logarithms and fundamental trigonometric identities to show that the pair of expressions is equivalent. In |csc 0+1|-2 In |cot 0 and - In |csc 0-1|
Use properties of natural logarithms and fundamental trigonometric identities to show that the pair of expressions is equivalent. In |csc 0+1|-2 In |cot 0 and - In |csc 0-1|
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Demonstrating Logarithm Properties and Trigonometric Identities
#### Objective:
Utilize properties of natural logarithms and fundamental trigonometric identities to establish the equivalence of the two given expressions.
#### Expressions:
\[ \ln | \csc \theta + 1 | - 2 \ln | \cot \theta | \quad \text{and} \quad - \ln | \csc \theta - 1 | \]
---
#### Step 1: Rewrite the Second Term Using the Power Rule of Logarithms
Given:
\[ \ln | \csc \theta + 1 | - 2 \ln | \cot \theta | = \ln | \csc \theta + 1 | - \ln \left( | \cot \theta |^2 \right) \]
Using the power rule of logarithms \( \ln(a^b) = b \ln(a) \), we rewrite:
\[ 2 \ln | \cot \theta | \quad \text{as} \quad \ln \left( | \cot \theta |^2 \right) \]
Thus, we obtain:
\[ \ln | \csc \theta + 1 | - \ln | \cot^2 \theta | \]
Finally, utilizing the properties of logarithms \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \), rewrite the expression:
\[ \ln \left( \frac{| \csc \theta + 1 |}{| \cot^2 \theta |} \right) \]
We've now simplified the expression:
\[ \ln | \csc \theta + 1 | - 2 \ln | \cot \theta | = \ln \left( \frac{| \csc \theta + 1 |}{| \cot^2 \theta |} \right) \]
This concludes our step of rewriting the second term using the power rule of logarithms.
---
#### Summary:
We demonstrated how logarithm properties and trigonometric identities are applied to express given mathematical terms equivalently. By leveraging the power rule and properties of logarithms, we simplified the initial expression into a form conducive to proving equivalence with the second expression.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3988bcd1-9898-42cb-8768-27ebf2c36c6f%2Fb8e2a61e-b44b-4e92-bf2f-372b0a17402e%2Fioxwjl_processed.png&w=3840&q=75)
Transcribed Image Text:### Demonstrating Logarithm Properties and Trigonometric Identities
#### Objective:
Utilize properties of natural logarithms and fundamental trigonometric identities to establish the equivalence of the two given expressions.
#### Expressions:
\[ \ln | \csc \theta + 1 | - 2 \ln | \cot \theta | \quad \text{and} \quad - \ln | \csc \theta - 1 | \]
---
#### Step 1: Rewrite the Second Term Using the Power Rule of Logarithms
Given:
\[ \ln | \csc \theta + 1 | - 2 \ln | \cot \theta | = \ln | \csc \theta + 1 | - \ln \left( | \cot \theta |^2 \right) \]
Using the power rule of logarithms \( \ln(a^b) = b \ln(a) \), we rewrite:
\[ 2 \ln | \cot \theta | \quad \text{as} \quad \ln \left( | \cot \theta |^2 \right) \]
Thus, we obtain:
\[ \ln | \csc \theta + 1 | - \ln | \cot^2 \theta | \]
Finally, utilizing the properties of logarithms \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \), rewrite the expression:
\[ \ln \left( \frac{| \csc \theta + 1 |}{| \cot^2 \theta |} \right) \]
We've now simplified the expression:
\[ \ln | \csc \theta + 1 | - 2 \ln | \cot \theta | = \ln \left( \frac{| \csc \theta + 1 |}{| \cot^2 \theta |} \right) \]
This concludes our step of rewriting the second term using the power rule of logarithms.
---
#### Summary:
We demonstrated how logarithm properties and trigonometric identities are applied to express given mathematical terms equivalently. By leveraging the power rule and properties of logarithms, we simplified the initial expression into a form conducive to proving equivalence with the second expression.
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