9. Establish the identity 1- sec 0 tan 0 2 cot 0 tan 0 1- sec 0
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![### Trigonometric Identity Problem
#### Problem 9: Establish the Identity
\[ \frac{1 - \sec \theta}{\tan \theta} - \frac{\tan \theta}{1 - \sec \theta} = 2 \cot \theta \]
This trigonometric identity requires you to transform and simplify the left-hand side of the equation to demonstrate that it is equal to the right-hand side, \(2 \cot \theta\).
#### Steps:
- **Left-hand Side:**
- Begin with \(\frac{1 - \sec \theta}{\tan \theta} - \frac{\tan \theta}{1 - \sec \theta}\).
- Use trigonometric identities such as \( \sec \theta = \frac{1}{\cos \theta} \) and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
- Simplify the expression by finding a common denominator or using algebraic manipulations.
- **Right-hand Side:**
- Recognize that \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \).
- The expression simplifies to \(2 \cdot \frac{\cos \theta}{\sin \theta}\).
By equating both sides through algebraic manipulation and simplification, the identity can be validated.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb5b35509-0b77-4a0a-88c0-a637e9e03cb4%2F46d043e7-ed34-41a0-b89f-30e402d6b2e8%2F0wgs9aq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Trigonometric Identity Problem
#### Problem 9: Establish the Identity
\[ \frac{1 - \sec \theta}{\tan \theta} - \frac{\tan \theta}{1 - \sec \theta} = 2 \cot \theta \]
This trigonometric identity requires you to transform and simplify the left-hand side of the equation to demonstrate that it is equal to the right-hand side, \(2 \cot \theta\).
#### Steps:
- **Left-hand Side:**
- Begin with \(\frac{1 - \sec \theta}{\tan \theta} - \frac{\tan \theta}{1 - \sec \theta}\).
- Use trigonometric identities such as \( \sec \theta = \frac{1}{\cos \theta} \) and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
- Simplify the expression by finding a common denominator or using algebraic manipulations.
- **Right-hand Side:**
- Recognize that \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \).
- The expression simplifies to \(2 \cdot \frac{\cos \theta}{\sin \theta}\).
By equating both sides through algebraic manipulation and simplification, the identity can be validated.
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