Ch. 5 - Trigonometric Identities

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

Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problem 5.RE.12

Chapter 6, Problem 5.RE.12

Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.
csc² θ + sec² θ

Verified step by step guidance

Recall the definitions of the cosecant and secant functions in terms of sine and cosine: \(\csc \theta = \frac{1}{\sin \theta}\) and \(\sec \theta = \frac{1}{\cos \theta}\).

Rewrite the given expression \(\csc^{2} \theta + \sec^{2} \theta\) using these definitions: \(\left(\frac{1}{\sin \theta}\right)^{2} + \left(\frac{1}{\cos \theta}\right)^{2}\).

Simplify the expression to \(\frac{1}{\sin^{2} \theta} + \frac{1}{\cos^{2} \theta}\), which is a sum of two fractions.

Find a common denominator for the two fractions, which is \(\sin^{2} \theta \cos^{2} \theta\), and rewrite the expression as \(\frac{\cos^{2} \theta}{\sin^{2} \theta \cos^{2} \theta} + \frac{\sin^{2} \theta}{\sin^{2} \theta \cos^{2} \theta}\).

Combine the fractions into a single fraction: \(\frac{\cos^{2} \theta + \sin^{2} \theta}{\sin^{2} \theta \cos^{2} \theta}\), then use the Pythagorean identity \(\sin^{2} \theta + \cos^{2} \theta = 1\) to simplify the numerator.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Reciprocal Trigonometric Identities

Reciprocal identities express functions like cosecant and secant in terms of sine and cosine: csc θ = 1/sin θ and sec θ = 1/cos θ. These allow rewriting expressions involving csc² θ and sec² θ into fractions with sin θ and cos θ in the denominator.

Pythagorean Identities

Pythagorean identities relate sine and cosine functions, such as sin² θ + cos² θ = 1. These identities help simplify expressions by replacing squared terms or combining fractions to eliminate quotients and reduce complexity.

Algebraic Simplification of Trigonometric Expressions

After substituting reciprocal identities, algebraic techniques like finding common denominators and combining fractions are used to simplify the expression. The goal is to write the expression without quotients and only in terms of sin θ and cos θ.

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Related Practice

Textbook Question

Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.

csc θ - sin θ

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Textbook Question

Use the given information to cos(x - y).

cos x = 2/9, sin y = -1/2, x in quadrant IV, y in quadrant III

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Textbook Question

Use the given information to find sin(x + y), cos(x - y), tan(x + y), and the quadrant of x + y.

sin x = 3/5, cos y = 24/25, x in quadrant I, y in quadrant IV

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Textbook Question

For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.

cos 75°

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Textbook Question

Use the given information to find each of the following.

sin A/2, given cos A/2 = - 3, 90° < A < 180°

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Textbook Question

Use the given information to find the quadrant of x + y.

sin y = - 2/3, cos x = -1/5 , x in quadrant II, y in quadrant III

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