Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
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![**Trigonometric Identity Proof**
**Problem Statement:**
Prove the identity:
\[ \sin(2\theta) = \frac{2\cot(\theta)}{1 + \cot^2(\theta)} \]
**Description:**
This problem requires you to prove a trigonometric identity involving the double angle formula for sine and the cotangent function. The expression on the right side involves cotangent, which is the reciprocal of tangent: \(\cot(\theta) = \frac{1}{\tan(\theta)}\).
The goal is to manipulate one side of the equation to match the other, using trigonometric identities such as:
- Double angle formulas:
- \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\)
- Relationship between sine, cosine, and cotangent:
- \(\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}\)
- Pythagorean identity:
- \(\sin^2(\theta) + \cos^2(\theta) = 1\)
You will likely need to express both sides in terms of sine and cosine, then simplify to demonstrate their equivalence.](https://content.bartleby.com/qna-images/question/53dc3663-7fd1-433e-97d7-8ab208c305a6/7d344796-e2f5-45cd-987f-aa9134fd0dc1/31o66u6_processed.jpeg)
Transcribed Image Text:**Trigonometric Identity Proof**
**Problem Statement:**
Prove the identity:
\[ \sin(2\theta) = \frac{2\cot(\theta)}{1 + \cot^2(\theta)} \]
**Description:**
This problem requires you to prove a trigonometric identity involving the double angle formula for sine and the cotangent function. The expression on the right side involves cotangent, which is the reciprocal of tangent: \(\cot(\theta) = \frac{1}{\tan(\theta)}\).
The goal is to manipulate one side of the equation to match the other, using trigonometric identities such as:
- Double angle formulas:
- \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\)
- Relationship between sine, cosine, and cotangent:
- \(\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}\)
- Pythagorean identity:
- \(\sin^2(\theta) + \cos^2(\theta) = 1\)
You will likely need to express both sides in terms of sine and cosine, then simplify to demonstrate their equivalence.
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