Verify the identity. (Simplify at each step.) 5 sin2 α-5 sint α-5 cos? α-5cos*α 5 sin? a – 5 sin“ a = 5 sin? a(1 – - (5-[ )(cos? a) = 5 cos? a – 5 cos“ a
Verify the identity. (Simplify at each step.) 5 sin2 α-5 sint α-5 cos? α-5cos*α 5 sin? a – 5 sin“ a = 5 sin? a(1 – - (5-[ )(cos? a) = 5 cos? a – 5 cos“ a
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter2: Right Triangle Trigonometry
Section: Chapter Questions
Problem 5GP
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![## Verification of Trigonometric Identity
### Problem Statement
Verify the following trigonometric identity. Simplify at each step to proceed.
\[ 5 \sin^2 \alpha - 5 \sin^4 \alpha = 5 \cos^2 \alpha - 5 \cos^4 \alpha \]
### Solution
1. **Original Expression:**
\[ 5 \sin^2 \alpha - 5 \sin^4 \alpha = 5 \cos^2 \alpha - 5 \cos^4 \alpha \]
2. **Factor out \(5 \sin^2 \alpha\) on the left-hand side:**
\[ 5 \sin^2 \alpha - 5 \sin^4 \alpha = 5 \sin^2 \alpha (1 - \sin^2 \alpha) \]
\[ \text{(Left side transformation)} \]
3. **Utilize the Pythagorean identity \(\sin^2 \alpha + \cos^2 \alpha = 1\):**
\[ = 5 \sin^2 \alpha (1 - \cos^2 \alpha) \]
\[ \text{(Replace \((1 - \sin^2 \alpha)\) with \(\cos^2 \alpha\))} \]
4. **Recognize that the factorization leads to an equivalent form on the right-hand side:**
\[ = (5 - 5 \cos^2 \alpha)(\cos^2 \alpha) \]
\[ \text{(Rearrangement)} \]
5. **Final simplified form:**
\[ = 5 \cos^2 \alpha - 5 \cos^4 \alpha \]
\[ \text{(Equivalence achieved)} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1f1fd054-0c44-4879-b08e-425a6935829a%2F05b7e1ca-adff-4e34-a3e2-c88397901969%2Fv43m9xj.png&w=3840&q=75)
Transcribed Image Text:## Verification of Trigonometric Identity
### Problem Statement
Verify the following trigonometric identity. Simplify at each step to proceed.
\[ 5 \sin^2 \alpha - 5 \sin^4 \alpha = 5 \cos^2 \alpha - 5 \cos^4 \alpha \]
### Solution
1. **Original Expression:**
\[ 5 \sin^2 \alpha - 5 \sin^4 \alpha = 5 \cos^2 \alpha - 5 \cos^4 \alpha \]
2. **Factor out \(5 \sin^2 \alpha\) on the left-hand side:**
\[ 5 \sin^2 \alpha - 5 \sin^4 \alpha = 5 \sin^2 \alpha (1 - \sin^2 \alpha) \]
\[ \text{(Left side transformation)} \]
3. **Utilize the Pythagorean identity \(\sin^2 \alpha + \cos^2 \alpha = 1\):**
\[ = 5 \sin^2 \alpha (1 - \cos^2 \alpha) \]
\[ \text{(Replace \((1 - \sin^2 \alpha)\) with \(\cos^2 \alpha\))} \]
4. **Recognize that the factorization leads to an equivalent form on the right-hand side:**
\[ = (5 - 5 \cos^2 \alpha)(\cos^2 \alpha) \]
\[ \text{(Rearrangement)} \]
5. **Final simplified form:**
\[ = 5 \cos^2 \alpha - 5 \cos^4 \alpha \]
\[ \text{(Equivalence achieved)} \]
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