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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![**Problem 6: Prove the Trigonometric Identity**
Prove that:
\[ \cos^4 x - \sin^4 x = \cos 2x \]
This is a trigonometric identity that demonstrates the relationship between the fourth powers of sine and cosine functions and the double angle cosine. To prove this identity, utilize trigonometric identities and algebraic manipulation.
**Solution Outline:**
1. Start with the left side of the equation, \( \cos^4 x - \sin^4 x \).
2. Use the difference of squares identity:
\[ a^2 - b^2 = (a - b)(a + b) \]
Thus, express \( \cos^4 x - \sin^4 x \) as:
\[ (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x) \]
3. Recall the Pythagorean identity:
\[ \cos^2 x + \sin^2 x = 1 \]
Substitute this into the expression:
\[ (\cos^2 x - \sin^2 x)(1) = \cos^2 x - \sin^2 x \]
4. Use the double angle identity for cosine:
\[ \cos 2x = \cos^2 x - \sin^2 x \]
Conclude that:
\[ \cos^4 x - \sin^4 x = \cos 2x \]
This demonstrates the correctness of the identity through algebraic manipulation and application of known trigonometric identities.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe708b20e-246d-43f9-aec0-b076b54d250e%2F93f55d8c-62ee-48df-8ea2-9c94d40447b8%2Fxiscdc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 6: Prove the Trigonometric Identity**
Prove that:
\[ \cos^4 x - \sin^4 x = \cos 2x \]
This is a trigonometric identity that demonstrates the relationship between the fourth powers of sine and cosine functions and the double angle cosine. To prove this identity, utilize trigonometric identities and algebraic manipulation.
**Solution Outline:**
1. Start with the left side of the equation, \( \cos^4 x - \sin^4 x \).
2. Use the difference of squares identity:
\[ a^2 - b^2 = (a - b)(a + b) \]
Thus, express \( \cos^4 x - \sin^4 x \) as:
\[ (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x) \]
3. Recall the Pythagorean identity:
\[ \cos^2 x + \sin^2 x = 1 \]
Substitute this into the expression:
\[ (\cos^2 x - \sin^2 x)(1) = \cos^2 x - \sin^2 x \]
4. Use the double angle identity for cosine:
\[ \cos 2x = \cos^2 x - \sin^2 x \]
Conclude that:
\[ \cos^4 x - \sin^4 x = \cos 2x \]
This demonstrates the correctness of the identity through algebraic manipulation and application of known trigonometric identities.
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