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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Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 6, Problem 5.72
Verify that each equation is an identity.
tan² α sin² α = tan² α + cos² α - 1
Verified step by step guidance1
Start by expressing \( \tan^2 \alpha \) in terms of \( \sin \alpha \) and \( \cos \alpha \). Recall that \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \), so \( \tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha} \).
Substitute \( \tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha} \) into the left side of the equation: \( \frac{\sin^2 \alpha}{\cos^2 \alpha} \sin^2 \alpha \).
Simplify the left side: \( \frac{\sin^4 \alpha}{\cos^2 \alpha} \).
Now, consider the right side of the equation: \( \tan^2 \alpha + \cos^2 \alpha - 1 \). Substitute \( \tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha} \) into the right side.
Simplify the right side: \( \frac{\sin^2 \alpha}{\cos^2 \alpha} + \cos^2 \alpha - 1 \). Use the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \) to simplify further and verify if both sides are equal.
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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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Fundamental Trigonometric Identities
Tangent and Sine Functions
The tangent function, defined as the ratio of sine to cosine (tan α = sin α / cos α), plays a key role in trigonometry. The sine function represents the ratio of the opposite side to the hypotenuse in a right triangle. Recognizing how these functions relate to each other is essential for manipulating and verifying trigonometric equations.
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5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²). This theorem underpins many trigonometric identities, particularly the relationship between sine, cosine, and tangent. It is fundamental for deriving and verifying identities involving these functions.
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5:19Solving Right Triangles with the Pythagorean Theorem
Related Practice
Textbook Question
Verify that each equation is an identity.
sin² θ (1 + cot² θ) - 1 = 0
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Textbook Question
Factor each trigonometric expression.
sin³ α + cos³ α
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Textbook Question
Simplify each expression.
±√[(1 - cos 8θ)/(1 + cos 8θ)]
768
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Textbook Question
Perform each indicated operation and simplify the result so that there are no quotients.
(1 + tan θ)² - 2 tan θ
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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.
-sec² (-θ) + sin² (-θ) + cos² (-θ)
759
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