Textbook Question
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
[1 - sin²(-θ)]/[1 + cot²(-θ)]
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.68
Textbook Question
Verify that each equation is an identity.
(sin⁴ α - cos⁴ α )/(sin² α - cos² α) = 1
Verified step by step guidance1
Recognize that the expression \( \sin^4 \alpha - \cos^4 \alpha \) is a difference of squares, which can be factored as \((\sin^2 \alpha - \cos^2 \alpha)(\sin^2 \alpha + \cos^2 \alpha)\).
Recall the Pythagorean identity: \( \sin^2 \alpha + \cos^2 \alpha = 1 \).
Substitute the identity into the factored expression: \((\sin^2 \alpha - \cos^2 \alpha)(1)\).
Notice that the denominator of the original expression is \( \sin^2 \alpha - \cos^2 \alpha \).
Cancel the common factor \( \sin^2 \alpha - \cos^2 \alpha \) from the numerator and the denominator, leaving \( 1 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations that hold true for all values of the variable within a certain domain. Common identities include the Pythagorean identities, reciprocal identities, and quotient identities. Understanding these identities is crucial for verifying equations and simplifying expressions in trigonometry.
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Difference of Squares
The difference of squares is a mathematical identity that states a² - b² = (a - b)(a + b). This concept is essential for factoring expressions involving squares, such as sin² α and cos² α. In the context of the given equation, recognizing this identity can simplify the verification process.
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Simplifying Rational Expressions
Simplifying rational expressions involves reducing fractions to their simplest form by canceling common factors. In trigonometry, this often requires applying identities and algebraic techniques. For the given equation, simplifying the left-hand side will help confirm whether it equals the right-hand side, which is a fundamental skill in solving trigonometric equations.
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