Consider the following. cos(a + b) cos(a) cos(b) Prove the identity. cos(a + b) cos(a) cos(b) = 1 tan(a) tan(b) = cos(a) cos(b) - cos(a) cos(b) cos(a) cos(b) "I cos(a) cos(b) cos(a) cos(b) = 1- 3).([ sin(a) cos(a) X
Consider the following. cos(a + b) cos(a) cos(b) Prove the identity. cos(a + b) cos(a) cos(b) = 1 tan(a) tan(b) = cos(a) cos(b) - cos(a) cos(b) cos(a) cos(b) "I cos(a) cos(b) cos(a) cos(b) = 1- 3).([ sin(a) cos(a) X
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
![### Transcription and Explanation
#### Problem Statement
Consider the following identity:
\[
\frac{\cos(a + b)}{\cos(a) \cos(b)} = 1 - \tan(a) \tan(b)
\]
Prove the identity.
---
#### Proof Structure
1. **Expression Setup**
\[
\frac{\cos(a + b)}{\cos(a) \cos(b)} = \frac{\cos(a) \cos(b) - \boxed{\cos(a)\cos(b)}}{\cos(a) \cos(b)}
\]
- A box is drawn around the term \(\cos(a)\cos(b)\) to likely indicate the part to be expanded or subtracted.
2. **Simplification**
\[
= \frac{\cos(a) \cos(b)}{\cos(a) \cos(b)} - \boxed{\cdots}
\]
3. **Further Simplification**
\[
= 1 - \left(\frac{\sin(a)}{\cos(a)}\right) \cdot \left( \boxed{\cdots} \right)
\]
4. **Final Result**
\[
= \boxed{\cdots}
\]
---
The proof involves expressing \(\cos(a + b)\) using trigonometric identities and simplifying it by subtracting terms and expanding using tangent and sine functions. The missing expressions, indicated by boxes, need to be filled with appropriate trigonometric simplifications to complete the proof.
This step-by-step decomposition highlights the key parts of the identity proof without revealing the detailed intermediate steps that should follow from trigonometric identities.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8e308213-df48-4360-8e9d-bfd0a573e9aa%2Fadf34214-8029-43b4-92e1-7510c45b7768%2F93d7zb_processed.png&w=3840&q=75)
Transcribed Image Text:### Transcription and Explanation
#### Problem Statement
Consider the following identity:
\[
\frac{\cos(a + b)}{\cos(a) \cos(b)} = 1 - \tan(a) \tan(b)
\]
Prove the identity.
---
#### Proof Structure
1. **Expression Setup**
\[
\frac{\cos(a + b)}{\cos(a) \cos(b)} = \frac{\cos(a) \cos(b) - \boxed{\cos(a)\cos(b)}}{\cos(a) \cos(b)}
\]
- A box is drawn around the term \(\cos(a)\cos(b)\) to likely indicate the part to be expanded or subtracted.
2. **Simplification**
\[
= \frac{\cos(a) \cos(b)}{\cos(a) \cos(b)} - \boxed{\cdots}
\]
3. **Further Simplification**
\[
= 1 - \left(\frac{\sin(a)}{\cos(a)}\right) \cdot \left( \boxed{\cdots} \right)
\]
4. **Final Result**
\[
= \boxed{\cdots}
\]
---
The proof involves expressing \(\cos(a + b)\) using trigonometric identities and simplifying it by subtracting terms and expanding using tangent and sine functions. The missing expressions, indicated by boxes, need to be filled with appropriate trigonometric simplifications to complete the proof.
This step-by-step decomposition highlights the key parts of the identity proof without revealing the detailed intermediate steps that should follow from trigonometric identities.
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