Factor the expression. Use the fundamental identities to simplify, if necessary. (There is more than one correct form of each answer.) 3 sec³(x) - 3 sec²(x) − 3 sec(x) + 3
Factor the expression. Use the fundamental identities to simplify, if necessary. (There is more than one correct form of each answer.) 3 sec³(x) - 3 sec²(x) − 3 sec(x) + 3
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.2: Trigonometric Functions Of Angles
Problem 82E
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![### Algebraic Factorization Exercise
#### Topic: Factorization of Trigonometric Expressions using Fundamental Identities
**Problem Statement:**
Factor the given trigonometric expression. Use fundamental identities to simplify it if necessary. Note that there could be more than one correct way to represent the simplified form.
\[ 3 \sec^3(x) - 3 \sec^2(x) - 3 \sec(x) + 3 \]
**Instructions:**
1. Identify common factors across the terms.
2. Apply fundamental trigonometric identities as needed to further simplify the expression.
3. Provide all possible simplified forms, if multiple answers exist.
**Expression to Factor:**
\[ 3 \sec^3(x) - 3 \sec^2(x) - 3 \sec(x) + 3 \]
**Solution Approach:**
1. **Factor Out the Common Term:**
Notice that each term has a common factor of 3:
\[ 3 (\sec^3(x) - \sec^2(x) - \sec(x) + 1) \]
2. **Group Terms:**
Group the terms to identify a common factor within the grouped terms:
\[ 3 \left[ (\sec^3(x) - \sec^2(x)) + (-\sec(x) + 1) \right] \]
3. **Factor Each Group:**
- From \(\sec^3(x) - \sec^2(x)\), factor out \(\sec^2(x)\):
\[ \sec^2(x) (\sec(x) - 1) \]
- The second group \(-\sec(x) + 1\) can be rewritten to show a common factor:
\[ -1 (\sec(x) - 1) \]
Therefore, the expression becomes:
\[ 3 \left[ \sec^2(x) (\sec(x) - 1) - 1 (\sec(x) - 1) \right] \]
4. **Factor the Binomial Common Factor:**
Recognize the common binomial factor \((\sec(x) - 1)\):
\[ 3 (\sec(x) - 1) [\sec^2(x) - 1] \]
5. **Simplify Using Trigonometric Identities:**
Recognize the Pythagorean](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0e38307-1ade-44bc-b712-aaeda4c58098%2Faa64fb9e-4cc2-49b4-a562-fc5c7bc85958%2Fiv1eq7i_processed.png&w=3840&q=75)
Transcribed Image Text:### Algebraic Factorization Exercise
#### Topic: Factorization of Trigonometric Expressions using Fundamental Identities
**Problem Statement:**
Factor the given trigonometric expression. Use fundamental identities to simplify it if necessary. Note that there could be more than one correct way to represent the simplified form.
\[ 3 \sec^3(x) - 3 \sec^2(x) - 3 \sec(x) + 3 \]
**Instructions:**
1. Identify common factors across the terms.
2. Apply fundamental trigonometric identities as needed to further simplify the expression.
3. Provide all possible simplified forms, if multiple answers exist.
**Expression to Factor:**
\[ 3 \sec^3(x) - 3 \sec^2(x) - 3 \sec(x) + 3 \]
**Solution Approach:**
1. **Factor Out the Common Term:**
Notice that each term has a common factor of 3:
\[ 3 (\sec^3(x) - \sec^2(x) - \sec(x) + 1) \]
2. **Group Terms:**
Group the terms to identify a common factor within the grouped terms:
\[ 3 \left[ (\sec^3(x) - \sec^2(x)) + (-\sec(x) + 1) \right] \]
3. **Factor Each Group:**
- From \(\sec^3(x) - \sec^2(x)\), factor out \(\sec^2(x)\):
\[ \sec^2(x) (\sec(x) - 1) \]
- The second group \(-\sec(x) + 1\) can be rewritten to show a common factor:
\[ -1 (\sec(x) - 1) \]
Therefore, the expression becomes:
\[ 3 \left[ \sec^2(x) (\sec(x) - 1) - 1 (\sec(x) - 1) \right] \]
4. **Factor the Binomial Common Factor:**
Recognize the common binomial factor \((\sec(x) - 1)\):
\[ 3 (\sec(x) - 1) [\sec^2(x) - 1] \]
5. **Simplify Using Trigonometric Identities:**
Recognize the Pythagorean
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