Add the two fractions and simplify if possible. Leave your answer in terms of sine and/or cose cos 8 sine + sine cos 8 cos 8+ sin 8 cos sin 0 1 cos 8 sin 8 cosesine sin 8 cos 8
Add the two fractions and simplify if possible. Leave your answer in terms of sine and/or cose cos 8 sine + sine cos 8 cos 8+ sin 8 cos sin 0 1 cos 8 sin 8 cosesine sin 8 cos 8
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Topic: Simplifying Trigonometric Expressions**
### Problem Statement:
Add the two fractions and simplify if possible. Leave your answer in terms of \( \sin \theta \) and/or \( \cos \theta \).
\[ \frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta} \]
### Multiple Choice Options:
1. \( \frac{\cos \theta + \sin \theta}{\cos \theta \sin \theta} \)
2. \( 1 \)
3. \( \cos \theta \sin \theta \)
4. \( \sin \theta \cos\theta \)
5. \( \frac{\sin \theta}{\cos \theta} \)
### Explanation:
To add the two fractions \(\frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta} \), we need to find a common denominator.
1. The common denominator of \(\sin \theta\) and \(\cos \theta\) is \( \sin \theta \cos \theta \).
2. Rewriting each fraction with this common denominator:
\[
\frac{\cos \theta \cdot \cos \theta}{\sin \theta \cos \theta} + \frac{\sin \theta \cdot \sin \theta}{\sin \theta \cos \theta}
\]
This simplifies to:
\[
\frac{\cos^2 \theta + \sin^2 \theta}{\sin \theta \cos \theta}
\]
3. Using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[
\frac{1}{\sin \theta \cos \theta}
\]
Thus, the simplified form of the given problem is \( \frac{1}{\sin \theta \cos \theta} \).
Therefore, the correct answer is:
- Option 2: \( 1 \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcccb93a5-bda7-4c84-b750-51d9219067a4%2Fddaaf7bc-2dbc-44dc-84d8-30df5e99eebc%2Fadkhhp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Simplifying Trigonometric Expressions**
### Problem Statement:
Add the two fractions and simplify if possible. Leave your answer in terms of \( \sin \theta \) and/or \( \cos \theta \).
\[ \frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta} \]
### Multiple Choice Options:
1. \( \frac{\cos \theta + \sin \theta}{\cos \theta \sin \theta} \)
2. \( 1 \)
3. \( \cos \theta \sin \theta \)
4. \( \sin \theta \cos\theta \)
5. \( \frac{\sin \theta}{\cos \theta} \)
### Explanation:
To add the two fractions \(\frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta} \), we need to find a common denominator.
1. The common denominator of \(\sin \theta\) and \(\cos \theta\) is \( \sin \theta \cos \theta \).
2. Rewriting each fraction with this common denominator:
\[
\frac{\cos \theta \cdot \cos \theta}{\sin \theta \cos \theta} + \frac{\sin \theta \cdot \sin \theta}{\sin \theta \cos \theta}
\]
This simplifies to:
\[
\frac{\cos^2 \theta + \sin^2 \theta}{\sin \theta \cos \theta}
\]
3. Using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[
\frac{1}{\sin \theta \cos \theta}
\]
Thus, the simplified form of the given problem is \( \frac{1}{\sin \theta \cos \theta} \).
Therefore, the correct answer is:
- Option 2: \( 1 \)
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