Prove the identity. sec (-x) – sin(-x) tan(-x) = cosx
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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![## Prove the Identity
Given the trigonometric identity to prove:
\[ \sec(-x) - \sin(-x) \tan(-x) = \cos x \]
### Note
Each statement must be based on a rule chosen from the Rule menu. To see a detailed description of a rule, select the More Information Button to the right of the rule.
### Steps and Corresponding Rules
| Statement | Rule |
|------------------------------------------------|-----------|
| \(\sec(-x) - \sin(-x) \tan(-x)\) | |
| \(\sec(-x) - \sin(-x) \tan(-x)\) | Algebra |
| \(\sec x - \sin(-x) \tan(-x)\) | Odd/Even |
| \(\sec x - (-\sin x) \tan(-x)\) | Odd/Even |
| \(\sec x - (-\sin x) (-\tan x)\) | Odd/Even |
| \(\sec x - \sin x \tan x\) | Simplify |
| \(\boxed{\cos x}\) | Identity |
### Explanation of the Rules:
- **Algebra**: Simplifying the expression.
- **Odd/Even**: Applying properties of sine, cosine, and tangent for negative angles.
- **Simplify**: Reducing the expression to its simplest form.
- **Identity**: Using a fundamental trigonometric identity to conclude the proof.
### Diagram Explanation
To the right of the rule table, there is a selection menu which allows for the application of various trigonometric rules:
- The vertical scroll list includes different trigonometric rules and identities.
- Symbols such as \(\cos\), \(\sin\), \(\tan\), \(\cot\), \(\sec\), \(\csc\), \(\pi\), parentheses, and square roots allow for selecting specific trigonometric functions and constants.
- Additional buttons include a help icon (?), a reset button (↻), and a validation button (X).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4c3dcdda-9290-4eb4-ba73-ee9de7b78657%2F097cd4fa-f40a-4567-afa7-0894084883a3%2Fv5efk76.png&w=3840&q=75)
Transcribed Image Text:## Prove the Identity
Given the trigonometric identity to prove:
\[ \sec(-x) - \sin(-x) \tan(-x) = \cos x \]
### Note
Each statement must be based on a rule chosen from the Rule menu. To see a detailed description of a rule, select the More Information Button to the right of the rule.
### Steps and Corresponding Rules
| Statement | Rule |
|------------------------------------------------|-----------|
| \(\sec(-x) - \sin(-x) \tan(-x)\) | |
| \(\sec(-x) - \sin(-x) \tan(-x)\) | Algebra |
| \(\sec x - \sin(-x) \tan(-x)\) | Odd/Even |
| \(\sec x - (-\sin x) \tan(-x)\) | Odd/Even |
| \(\sec x - (-\sin x) (-\tan x)\) | Odd/Even |
| \(\sec x - \sin x \tan x\) | Simplify |
| \(\boxed{\cos x}\) | Identity |
### Explanation of the Rules:
- **Algebra**: Simplifying the expression.
- **Odd/Even**: Applying properties of sine, cosine, and tangent for negative angles.
- **Simplify**: Reducing the expression to its simplest form.
- **Identity**: Using a fundamental trigonometric identity to conclude the proof.
### Diagram Explanation
To the right of the rule table, there is a selection menu which allows for the application of various trigonometric rules:
- The vertical scroll list includes different trigonometric rules and identities.
- Symbols such as \(\cos\), \(\sin\), \(\tan\), \(\cot\), \(\sec\), \(\csc\), \(\pi\), parentheses, and square roots allow for selecting specific trigonometric functions and constants.
- Additional buttons include a help icon (?), a reset button (↻), and a validation button (X).
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