Textbook Question
Verify that each equation is an identity.
2 cos³ x - cos x = (cos² x - sin² x)/sec x
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.54
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
Verified step by step guidance1
Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
Substitute \( \tan \theta \) with \( \frac{\sin \theta}{\cos \theta} \) in the expression \( \tan \theta \cos \theta \).
The expression becomes \( \frac{\sin \theta}{\cos \theta} \cdot \cos \theta \).
Simplify the expression by canceling out \( \cos \theta \) in the numerator and denominator.
The simplified expression is \( \sin \theta \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Functions
Trigonometric functions, including sine (sin), cosine (cos), and tangent (tan), are fundamental in trigonometry. The tangent function is defined as the ratio of the sine and cosine functions: tan(θ) = sin(θ) / cos(θ). Understanding these functions and their relationships is essential for manipulating and simplifying trigonometric expressions.
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Quotient Identity
The quotient identity in trigonometry states that the tangent of an angle can be expressed as the ratio of the sine and cosine of that angle. This identity is crucial for rewriting expressions involving tangent in terms of sine and cosine, allowing for simplification and elimination of quotients in the expression.
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Simplification of Trigonometric Expressions
Simplifying trigonometric expressions involves rewriting them in a more manageable form, often eliminating fractions and combining like terms. This process typically requires the use of identities, such as the Pythagorean identities and the quotient identities, to express all functions in terms of sine and cosine, which is a common requirement in trigonometric problems.
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