Prove the identity. sin(x + y) sin(x - y) = 2 cos(x) sin(y) Use the Sum and Difference Identities for Sine, and then simplify. sin(x + y) sin(x - y) - Read It Need Help? = sin(x) cos(y) + cos(x) sin(y) - (sin(x) cos(y) Watch It
Prove the identity. sin(x + y) sin(x - y) = 2 cos(x) sin(y) Use the Sum and Difference Identities for Sine, and then simplify. sin(x + y) sin(x - y) - Read It Need Help? = sin(x) cos(y) + cos(x) sin(y) - (sin(x) cos(y) Watch It
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
![### Proving Trigonometric Identities
In this lesson, we will prove the following trigonometric identity using the Sum and Difference Identities for Sine:
\[ \sin(x + y) - \sin(x - y) = 2 \cos(x) \sin(y) \]
### Steps to Prove the Identity
1. **Apply the Sum and Difference Identities for Sine:**
\[
\sin(x + y) - \sin(x - y) = \sin(x) \cos(y) + \cos(x) \sin(y) - \left( \sin(x) \cos(y) - \cos(x) \sin(y) \right)
\]
2. **Simplify the Expression:**
\[
\sin(x) \cos(y) + \cos(x) \sin(y) - \left( \sin(x) \cos(y) - \cos(x) \sin(y) \right)
\]
Breaking this down:
\[
= \sin(x) \cos(y) + \cos(x) \sin(y) - \sin(x) \cos(y) + \cos(x) \sin(y)
\]
3. **Combine Like Terms:**
\[
= \left( \sin(x) \cos(y) - \sin(x) \cos(y) \right) + \left( \cos(x) \sin(y) + \cos(x) \sin(y) \right)
\]
\[
= 0 + 2 \cos(x) \sin(y)
\]
4. **Final Simplified Form:**
\[
\sin(x + y) - \sin(x - y) = 2 \cos(x) \sin(y)
\]
Thus, we have successfully proven the identity.
### Need Help?
If you need further assistance understanding this content, you can:
- **Read It**: Click here to access detailed written explanations for proving trigonometric identities.
- **Watch It**: Click here to watch a video tutorial on the topic.
---
This concludes the proof for the given trigonometric identity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6841f1bf-ebcf-44b9-bde6-b1a56299f544%2F46d506ea-ecc8-4fc3-98b1-bc24c79c393b%2Fanme4cg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Proving Trigonometric Identities
In this lesson, we will prove the following trigonometric identity using the Sum and Difference Identities for Sine:
\[ \sin(x + y) - \sin(x - y) = 2 \cos(x) \sin(y) \]
### Steps to Prove the Identity
1. **Apply the Sum and Difference Identities for Sine:**
\[
\sin(x + y) - \sin(x - y) = \sin(x) \cos(y) + \cos(x) \sin(y) - \left( \sin(x) \cos(y) - \cos(x) \sin(y) \right)
\]
2. **Simplify the Expression:**
\[
\sin(x) \cos(y) + \cos(x) \sin(y) - \left( \sin(x) \cos(y) - \cos(x) \sin(y) \right)
\]
Breaking this down:
\[
= \sin(x) \cos(y) + \cos(x) \sin(y) - \sin(x) \cos(y) + \cos(x) \sin(y)
\]
3. **Combine Like Terms:**
\[
= \left( \sin(x) \cos(y) - \sin(x) \cos(y) \right) + \left( \cos(x) \sin(y) + \cos(x) \sin(y) \right)
\]
\[
= 0 + 2 \cos(x) \sin(y)
\]
4. **Final Simplified Form:**
\[
\sin(x + y) - \sin(x - y) = 2 \cos(x) \sin(y)
\]
Thus, we have successfully proven the identity.
### Need Help?
If you need further assistance understanding this content, you can:
- **Read It**: Click here to access detailed written explanations for proving trigonometric identities.
- **Watch It**: Click here to watch a video tutorial on the topic.
---
This concludes the proof for the given trigonometric identity.
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