Ch. 5 - Trigonometric Identities

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

Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problem 71

Chapter 6, Problem 71

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)
cos 2x = 1 - 2 sin² x

Verified step by step guidance

Start by expressing \( \cos 2x \) using the angle addition formula for cosine: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). Since \( 2x = x + x \), write \( \cos 2x = \cos x \cos x - \sin x \sin x \).

Simplify the expression to get \( \cos 2x = \cos^2 x - \sin^2 x \). This is one of the standard double-angle formulas for cosine.

Recall the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \). Use this to rewrite \( \cos^2 x \) in terms of \( \sin^2 x \): \( \cos^2 x = 1 - \sin^2 x \).

Substitute \( \cos^2 x = 1 - \sin^2 x \) into the expression \( \cos 2x = \cos^2 x - \sin^2 x \) to get \( \cos 2x = (1 - \sin^2 x) - \sin^2 x \).

Simplify the right-hand side to obtain \( \cos 2x = 1 - 2 \sin^2 x \), which matches the given identity, thus verifying it.

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Key Concepts

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

Double-Angle Identity for Cosine

The double-angle identity expresses cos 2x in terms of functions of x. It can be written as cos 2x = cos² x - sin² x, or equivalently as cos 2x = 1 - 2 sin² x. This identity helps simplify expressions involving angles multiplied by two.

Pythagorean Identity

The Pythagorean identity states that sin² x + cos² x = 1 for any angle x. This fundamental relationship allows substitution between sine and cosine squared terms, which is essential when verifying or transforming trigonometric identities.

Trigonometric Identity Verification

Verifying an identity involves showing that both sides of an equation are equivalent for all values of the variable. This often requires algebraic manipulation using known identities, such as the double-angle and Pythagorean identities, to rewrite one side to match the other.

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Fundamental Trigonometric Identities

Related Practice

Textbook Question

Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

csc 72°

Textbook Question

Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

tan 72°

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Textbook Question

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)

cos 2x = (cot² x - 1)/(cot² x + 1)

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Textbook Question

Verify that each equation is an identity.

(tan(α + β) - tan β)/(1 + tan(α + β) tan β) = tan α

Textbook Question

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)

cos 2x = cos² x - sin² x