Textbook Question
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
1 - 1/sec² x
6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.48
Textbook Question
Verify that each equation is an identity.
(tan² α + 1)/ sec α = sec α
Verified step by step guidance1
Start by recalling the Pythagorean identity: \( \tan^2 \alpha + 1 = \sec^2 \alpha \).
Substitute \( \sec^2 \alpha \) for \( \tan^2 \alpha + 1 \) in the left-hand side of the equation, giving \( \frac{\sec^2 \alpha}{\sec \alpha} \).
Simplify the expression \( \frac{\sec^2 \alpha}{\sec \alpha} \) by canceling one \( \sec \alpha \) from the numerator and the denominator, resulting in \( \sec \alpha \).
Observe that the simplified left-hand side \( \sec \alpha \) is equal to the right-hand side \( \sec \alpha \).
Conclude that the original equation \( \frac{\tan^2 \alpha + 1}{\sec \alpha} = \sec \alpha \) is indeed an identity.
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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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Tangent and Secant Functions
The tangent function, tan(α), is defined as the ratio of the opposite side to the adjacent side in a right triangle, or as sin(α)/cos(α). The secant function, sec(α), is the reciprocal of the cosine function, defined as 1/cos(α). Recognizing the relationships between these functions is essential for manipulating and verifying trigonometric equations.
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Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. This includes factoring, expanding, and combining like terms. In the context of trigonometric identities, effective algebraic manipulation allows one to transform one side of an equation to match the other, thereby verifying the identity.
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