Textbook Question
Factor each trigonometric expression.
(tan x + cot x)² - (tan x - cot x)²
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 5.40
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
csc² t - 1
Verified step by step guidance1
Recognize that the expression \( \csc^2 t - 1 \) can be related to a fundamental trigonometric identity.
Recall the Pythagorean identity: \( \csc^2 t = 1 + \cot^2 t \).
Substitute \( \csc^2 t \) in the expression with \( 1 + \cot^2 t \) to get \( (1 + \cot^2 t) - 1 \).
Simplify the expression by subtracting 1: \( \cot^2 t \).
Conclude that the simplified expression is \( \cot^2 t \), which is a power of a trigonometric function.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cosecant Function
The cosecant function, denoted as csc(t), is the reciprocal of the sine function. It is defined as csc(t) = 1/sin(t). Understanding this function is crucial for simplifying expressions involving csc²(t), as it relates directly to the sine function and its properties.
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Graphs of Secant and Cosecant Functions
Pythagorean Identity
The Pythagorean identities are fundamental relationships in trigonometry that relate the squares of the sine and cosine functions. One key identity is sin²(t) + cos²(t) = 1. This identity can be rearranged to express csc²(t) in terms of sin²(t), which is essential for simplifying expressions like csc²(t) - 1.
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6:25Pythagorean Identities
Fundamental Trigonometric Identities
Fundamental trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where the functions are defined. These include reciprocal identities, Pythagorean identities, and co-function identities. Utilizing these identities allows for the simplification of complex trigonometric expressions into more manageable forms.
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5:32Fundamental Trigonometric Identities
Related Practice
Textbook Question
Advanced methods of trigonometry can be used to find the following exact value.
sin 18° = (√5 - 1)/4
(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.
csc 18°
Textbook Question
Determine whether the positive or negative square root should be selected.
sin (-10°) = ± √[(1 - cos (-20°))/2]
1
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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.
tan(-θ)/sec θ
Textbook Question
Perform each indicated operation and simplify the result so that there are no quotients.
1/( sin α - 1) - 1/(sin α + 1)
Textbook Question
Find the remaining five trigonometric functions of θ.
sin θ = -4/5, cos θ < 0