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...
Question
**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: \( 1 \)
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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