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.
1 + cot(-θ)/cot(-θ)
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.66
Textbook Question
Verify that each equation is an identity.
sin² θ (1 + cot² θ) - 1 = 0
Verified step by step guidance1
Start by recalling the Pythagorean identity: \(1 + \cot^2 \theta = \csc^2 \theta\).
Substitute \(1 + \cot^2 \theta\) with \(\csc^2 \theta\) in the equation: \(\sin^2 \theta \cdot \csc^2 \theta - 1 = 0\).
Remember that \(\csc \theta = \frac{1}{\sin \theta}\), so \(\csc^2 \theta = \frac{1}{\sin^2 \theta}\).
Substitute \(\csc^2 \theta\) with \(\frac{1}{\sin^2 \theta}\) in the equation: \(\sin^2 \theta \cdot \frac{1}{\sin^2 \theta} - 1 = 0\).
Simplify the expression: \(1 - 1 = 0\), which confirms the identity.
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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 verifying equations, as they allow us to manipulate and simplify expressions to show equivalence.
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Cotangent Function
The cotangent function, denoted as cot(θ), is the reciprocal of the tangent function, defined as cot(θ) = cos(θ)/sin(θ). It can also be expressed in terms of sine and cosine, which is essential for transforming and simplifying trigonometric expressions. Recognizing how cotangent relates to sine and cosine is vital for verifying identities involving cotangent.
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Pythagorean Identity
The Pythagorean identity states that sin²(θ) + cos²(θ) = 1 for any angle θ. This fundamental identity is often used to simplify trigonometric expressions and verify identities. In the context of the given equation, recognizing how to apply this identity can help in transforming the left-hand side to demonstrate that it equals zero.
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