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 - 1/sec² x
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 5.80
Verify that each equation is an identity.
(1 + sin x + cos x)² = 2(1 + sin x) (1 + cos x)
Verified step by step guidance1
Start by expanding the left side of the equation: \((1 + \sin x + \cos x)^2\). Use the formula \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc\).
Calculate each term: \(a^2 = 1^2 = 1\), \(b^2 = (\sin x)^2 = \sin^2 x\), \(c^2 = (\cos x)^2 = \cos^2 x\), \(2ab = 2 \cdot 1 \cdot \sin x = 2\sin x\), \(2ac = 2 \cdot 1 \cdot \cos x = 2\cos x\), \(2bc = 2 \cdot \sin x \cdot \cos x = 2\sin x \cos x\).
Combine these terms to get: \(1 + \sin^2 x + \cos^2 x + 2\sin x + 2\cos x + 2\sin x \cos x\).
Use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) to simplify: \(1 + 1 + 2\sin x + 2\cos x + 2\sin x \cos x = 2 + 2\sin x + 2\cos x + 2\sin x \cos x\).
Now expand the right side: \(2(1 + \sin x)(1 + \cos x) = 2((1 + \sin x) + (1 + \sin x)\cos x) = 2(1 + \sin x + \cos x + \sin x \cos x)\). Simplify to see 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 within a certain domain. Common identities include the Pythagorean identities, reciprocal identities, and angle sum/difference identities. Understanding these identities is crucial for simplifying trigonometric expressions and verifying equations.
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Algebraic Expansion
Algebraic expansion involves applying the distributive property to multiply expressions, such as binomials. In the context of the given equation, expanding (1 + sin x + cos x)² requires using the formula (a + b)² = a² + 2ab + b², which helps in simplifying the left-hand side of the equation for comparison with the right-hand side.
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Factoring
Factoring is the process of breaking down an expression into simpler components, or factors, that can be multiplied to obtain the original expression. In the context of verifying identities, recognizing common factors on both sides of the equation can simplify the verification process and help establish equality between the two sides.
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Related Practice
Textbook Question
Verify that each equation is an identity.
sin² α + tan² α + cos² α = sec² α
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Textbook Question
Find sinθ.
sec θ = 7/2, tan θ < 0
Textbook Question
Verify that each equation is an identity.
sin θ + cos θ = sin θ/(1 - cot θ) + cos θ/(1 - tan θ)
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 cos θ
Textbook Question
Write each function as an expression involving functions of θ or x alone. See Example 2.
sin (3π/4 - x)
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