Ch. 5 - Trigonometric Identities

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problem 5.66

Chapter 6, Problem 5.66

Verify that each equation is an identity.
sin² θ (1 + cot² θ) - 1 = 0

Verified step by step guidance

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.

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.

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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Pythagorean Identities

Related Practice

Textbook Question

Verify that each equation is an identity.

(sin⁴ α - cos⁴ α )/(sin² α - cos² α) = 1

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.

-sec² (-θ) + sin² (-θ) + cos² (-θ)

Textbook Question

Match each expression in Column I with its value in Column II.

10. cos 67.5°

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Textbook Question

Verify that each equation is an identity.

sin θ/(1 - cos θ) - sin θ cos θ/( 1 + cos θ) = csc θ (1 + cos² θ)

Textbook Question

Perform each transformation. See Example 2.

Write cot x in terms of sin x.

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