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/ tan² α + cot α tan α
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.46
Textbook Question
Verify that each equation is an identity.
tan α/sec α = sin α
Verified step by step guidance1
Start by expressing \( \tan \alpha \) and \( \sec \alpha \) in terms of \( \sin \alpha \) and \( \cos \alpha \). Recall that \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and \( \sec \alpha = \frac{1}{\cos \alpha} \).
Substitute these expressions into the left side of the equation: \( \frac{\tan \alpha}{\sec \alpha} = \frac{\frac{\sin \alpha}{\cos \alpha}}{\frac{1}{\cos \alpha}} \).
Simplify the expression by multiplying the numerator by the reciprocal of the denominator: \( \frac{\sin \alpha}{\cos \alpha} \times \cos \alpha \).
Notice that the \( \cos \alpha \) terms cancel each other out, leaving \( \sin \alpha \).
Conclude that the left side simplifies to \( \sin \alpha \), which matches the right side of the equation, thus verifying the 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 trigonometric expressions.
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Tangent and Secant Functions
The tangent function, tan(α), is defined as the ratio of the sine and cosine functions: tan(α) = sin(α)/cos(α). The secant function, sec(α), is the reciprocal of the cosine function: sec(α) = 1/cos(α). Recognizing these relationships is essential for manipulating and verifying trigonometric equations.
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Sine Function
The sine function, sin(α), represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. It is a fundamental trigonometric function that plays a key role in various identities and equations. Understanding how sine interacts with other trigonometric functions is vital for verifying identities.
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