8. Prove the identity sec 0 – cos 0 = sin 0 tan 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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Question
![**Problem 8: Proving a Trigonometric Identity**
**Objective:**
Prove the trigonometric identity:
\[ \sec \theta - \cos \theta = \sin \theta \tan \theta \]
**Approach:**
To verify this identity, we will manipulate one side of the equation to transform it into the other side, using basic trigonometric identities and operations.
1. Recall the definitions:
- \( \sec \theta = \frac{1}{\cos \theta} \)
- \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
2. Start with the left side of the equation:
\[ \sec \theta - \cos \theta = \frac{1}{\cos \theta} - \cos \theta \]
3. Combine the terms over a common denominator:
\[ = \frac{1 - \cos^2 \theta}{\cos \theta} \]
4. Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to substitute \( 1 - \cos^2 \theta = \sin^2 \theta \):
\[ = \frac{\sin^2 \theta}{\cos \theta} \]
5. Split the fraction:
\[ = \sin \theta \cdot \frac{\sin \theta}{\cos \theta} \]
6. Simplify using the definition of tangent:
\[ = \sin \theta \cdot \tan \theta \]
Thus, we have shown that:
\[ \sec \theta - \cos \theta = \sin \theta \tan \theta \]
The identity is proven.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe5eb9b1d-c2a7-4d17-996f-fe9f500c8923%2Fe22600f8-5fc3-416b-82e7-16b529150da8%2Fima4f7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 8: Proving a Trigonometric Identity**
**Objective:**
Prove the trigonometric identity:
\[ \sec \theta - \cos \theta = \sin \theta \tan \theta \]
**Approach:**
To verify this identity, we will manipulate one side of the equation to transform it into the other side, using basic trigonometric identities and operations.
1. Recall the definitions:
- \( \sec \theta = \frac{1}{\cos \theta} \)
- \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
2. Start with the left side of the equation:
\[ \sec \theta - \cos \theta = \frac{1}{\cos \theta} - \cos \theta \]
3. Combine the terms over a common denominator:
\[ = \frac{1 - \cos^2 \theta}{\cos \theta} \]
4. Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to substitute \( 1 - \cos^2 \theta = \sin^2 \theta \):
\[ = \frac{\sin^2 \theta}{\cos \theta} \]
5. Split the fraction:
\[ = \sin \theta \cdot \frac{\sin \theta}{\cos \theta} \]
6. Simplify using the definition of tangent:
\[ = \sin \theta \cdot \tan \theta \]
Thus, we have shown that:
\[ \sec \theta - \cos \theta = \sin \theta \tan \theta \]
The identity is proven.
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