Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Question
**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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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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