Textbook Question
Factor each trigonometric expression.
sec² θ - 1
1
views
Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161
Not the one you use?Change textbookSelect a different textbook
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 5.70
Verify that each equation is an identity.
(cot² t - 1)/(1 + cot² t) = 1 - 2 sin² t
Verified step by step guidance1
Step 1: Recall the Pythagorean identity: \(1 + \cot^2 t = \csc^2 t\).
Step 2: Substitute \(\csc^2 t\) for \(1 + \cot^2 t\) in the left-hand side of the equation: \((\cot^2 t - 1)/\csc^2 t\).
Step 3: Use the identity \(\csc^2 t = 1/\sin^2 t\) to rewrite the expression: \((\cot^2 t - 1) \cdot \sin^2 t\).
Step 4: Recall that \(\cot^2 t = \cos^2 t/\sin^2 t\), and substitute it into the expression: \((\cos^2 t/\sin^2 t - 1) \cdot \sin^2 t\).
Step 5: Simplify the expression to show that it equals \(1 - 2\sin^2 t\), verifying the identity.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
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 expressions in trigonometry.
Recommended video:
5:32
Fundamental Trigonometric Identities
Cotangent Function
The cotangent function, denoted as cot(t), is the reciprocal of the tangent function, defined as cot(t) = cos(t)/sin(t). It can also be expressed in terms of sine and cosine, which is essential for manipulating and simplifying trigonometric expressions. Recognizing how cotangent relates to other trigonometric functions is key to solving the given equation.
Recommended video:
5:37Introduction to Cotangent Graph
Sine Function and Its Relationship to Cotangent
The sine function, sin(t), is a fundamental trigonometric function that relates to the cotangent function through the identity cot(t) = cos(t)/sin(t). The expression 1 - 2sin²(t) can be derived from the double angle formulas and is often used in conjunction with cotangent to verify identities. Understanding this relationship helps in transforming and equating both sides of the given equation.
Recommended video:
5:53Graph of Sine and Cosine Function
Related Practice
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.
(sin θ - cos θ) (csc θ + sec θ)
6
views
Textbook Question
For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.
cos² x
II
A. sin ^2 x/cos ^2 x
B.1/(sec ^2 x)
C. sin (-x)
D. csc ^2 x-cot ^2 x + sin ^2 x
E. tan x
Textbook Question
Verify that each equation is an identity.
(sin² θ)/cos θ = sec θ - cos θ
1
views
Textbook Question
Find sinθ.
csc θ = -8/5
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)