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.
tan θ cos θ
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.56
Textbook Question
Verify that each equation is an identity.
sec⁴ x - sec² x = tan⁴ x + tan² x
Verified step by step guidance1
Start by recalling the Pythagorean identity: \( \sec^2 x = 1 + \tan^2 x \).
Substitute \( \sec^2 x = 1 + \tan^2 x \) into the left side of the equation: \( \sec^4 x - \sec^2 x \).
Rewrite \( \sec^4 x \) as \( (\sec^2 x)^2 \) and substitute \( \sec^2 x = 1 + \tan^2 x \) to get \( ((1 + \tan^2 x)^2) - (1 + \tan^2 x) \).
Expand \( (1 + \tan^2 x)^2 \) to \( 1 + 2\tan^2 x + \tan^4 x \) and simplify the expression: \( 1 + 2\tan^2 x + \tan^4 x - 1 - \tan^2 x \).
Combine like terms to verify the identity: \( \tan^4 x + \tan^2 x \).
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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 where both sides are defined. 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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Secant and Tangent Functions
The secant function, sec(x), is defined as the reciprocal of the cosine function, while the tangent function, tan(x), is the ratio of the sine function to the cosine function. These functions are fundamental in trigonometry and are often used in identities and equations. Recognizing their relationships helps in manipulating and verifying trigonometric equations.
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Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations using algebraic rules. This includes factoring, expanding, and combining like terms. In the context of verifying trigonometric identities, effective algebraic manipulation allows one to transform one side of the equation to match the other, confirming the identity.
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