Textbook Question
Perform each transformation. See Example 2.
Write cot x in terms of csc x.
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.52
Textbook Question
Verify that each equation is an identity.
sin² α + tan² α + cos² α = sec² α
Verified step by step guidance1
Start by recalling the Pythagorean identity: \( \sin^2 \alpha + \cos^2 \alpha = 1 \).
Recall the identity for tangent and secant: \( \tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha} \) and \( \sec^2 \alpha = 1 + \tan^2 \alpha \).
Substitute \( \tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha} \) into the equation: \( \sin^2 \alpha + \frac{\sin^2 \alpha}{\cos^2 \alpha} + \cos^2 \alpha = \sec^2 \alpha \).
Combine the terms on the left side over a common denominator: \( \frac{\sin^2 \alpha \cdot \cos^2 \alpha + \sin^2 \alpha + \cos^4 \alpha}{\cos^2 \alpha} \).
Simplify the expression and verify if it equals \( \sec^2 \alpha \), using the identity \( \sec^2 \alpha = 1 + \tan^2 \alpha \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Pythagorean Identity
The Pythagorean identity states that for any angle α, the relationship sin² α + cos² α = 1 holds true. This fundamental identity is crucial in trigonometry as it connects the sine and cosine functions, allowing for simplifications and transformations in various equations.
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Definition of Tangent and Secant
Tangent and secant are defined in terms of sine and cosine: tan α = sin α / cos α and sec α = 1 / cos α. Understanding these definitions is essential for manipulating trigonometric identities, as they allow us to express equations in terms of sine and cosine, facilitating verification of identities.
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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. Recognizing and applying these identities is key to verifying equations and simplifying trigonometric expressions.
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