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.
(sec θ - 1) (sec θ + 1)
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 5.60
Verify that each equation is an identity.
[(sec θ - tan θ)² + 1]/(sec θ csc θ - tan θ csc θ) = 2 tan θ
Verified step by step guidance1
Step 1: Start by simplifying the left-hand side (LHS) of the equation. Expand \((\sec \theta - \tan \theta)^2\) using the identity \((a - b)^2 = a^2 - 2ab + b^2\).
Step 2: Use the Pythagorean identities \(\sec^2 \theta = 1 + \tan^2 \theta\) and \(\csc^2 \theta = 1 + \cot^2 \theta\) to simplify the expression further.
Step 3: Simplify the denominator \(\sec \theta \csc \theta - \tan \theta \csc \theta\) by expressing \(\sec \theta\) and \(\csc \theta\) in terms of sine and cosine: \(\sec \theta = \frac{1}{\cos \theta}\) and \(\csc \theta = \frac{1}{\sin \theta}\).
Step 4: Substitute the simplified expressions back into the LHS and simplify the fraction. Look for common terms that can be canceled out.
Step 5: Verify that the simplified LHS equals the right-hand side (RHS), \(2 \tan \theta\), by ensuring both sides are equivalent after simplification.
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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 simplifying trigonometric expressions and verifying equations as identities.
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Fundamental Trigonometric Identities
Secant and Tangent Functions
The secant function (sec θ) is the reciprocal of the cosine function, while the tangent function (tan θ) is the ratio of the sine function to the cosine function. These functions are fundamental in trigonometry and often appear in various identities and equations. Recognizing their relationships helps in manipulating and simplifying expressions involving them.
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Algebraic Manipulation in Trigonometry
Algebraic manipulation involves rearranging and simplifying expressions to prove identities. This includes factoring, expanding, and combining like terms. In trigonometry, it is essential to apply these techniques to transform one side of an equation into the other, thereby verifying the identity effectively.
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04:12Algebraic Operations on Vectors
Related Practice
Textbook Question
Verify that each equation is an identity.
(tan² α + 1)/ sec α = sec α
1
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Textbook Question
Match each expression in Column I with its value in Column II.
cos² (π/6) - sin² (π/6)
Textbook Question
Find the exact value of each expression. (Do not use a calculator.)
cos(-15°)
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.
cot² θ(1 + tan² θ)
Textbook Question
Verify that each equation is an identity.
(sec α - tan α)² = (1 - sin α)/(1 + sin α)
4
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