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.
csc θ cos θ tan θ
2
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 5.24
Factor each trigonometric expression.
sec² θ - 1
Verified step by step guidance1
Recognize that the expression \( \sec^2 \theta - 1 \) is a difference of squares.
Recall the Pythagorean identity: \( \sec^2 \theta = 1 + \tan^2 \theta \).
Substitute \( \sec^2 \theta \) with \( 1 + \tan^2 \theta \) in the expression: \( (1 + \tan^2 \theta) - 1 \).
Simplify the expression: \( \tan^2 \theta \).
Factor \( \tan^2 \theta \) as \( (\tan \theta)(\tan \theta) \).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Was this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. One of the fundamental identities is the Pythagorean identity, which states that sec² θ = 1 + tan² θ. Understanding these identities is crucial for simplifying and factoring trigonometric expressions.
Recommended video:
5:32
Fundamental Trigonometric Identities
Factoring Techniques
Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. Common techniques include recognizing patterns such as the difference of squares, which applies to expressions like sec² θ - 1. Mastery of these techniques is essential for solving trigonometric equations efficiently.
Recommended video:
6:08Factoring
Difference of Squares
The difference of squares is a specific algebraic identity that states a² - b² = (a - b)(a + b). This concept is particularly useful in factoring expressions where one term is the square of a variable or function. In the case of sec² θ - 1, it can be factored as (sec θ - 1)(sec θ + 1), illustrating the application of this identity in trigonometry.
Recommended video:
4:47Sum and Difference of Tangent
Related Practice
Textbook Question
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
cot t tan t
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
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)
Textbook Question
Verify that each equation is an identity.
sec⁴ x - sec² x = tan⁴ x + tan² x
Textbook Question
Perform each transformation. See Example 2.
Write cot x in terms of sin x.
4
views