Ch. 5 - Trigonometric Identities

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

Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problem 5.1.43

Chapter 6, Problem 5.1.43

For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.
cos² x

II
A. sin ^2 x/cos ^2 x
B.1/(sec ^2 x)
C. sin (-x)
D. csc ^2 x-cot ^2 x + sin ^2 x
E. tan x

Verified step by step guidance

Recall the Pythagorean identity in trigonometry: \(\cos^{2} x + \sin^{2} x = 1\).

From this identity, express \(\cos^{2} x\) in terms of \(\sin^{2} x\) by rearranging the equation: \(\cos^{2} x = 1 - \sin^{2} x\).

Recognize that sometimes expressions in Column II might be written differently, such as \(1 - \sin^{2} x\) or other equivalent forms.

Check if the expression in Column II matches \(1 - \sin^{2} x\) or can be rewritten to this form to complete the identity with \(\cos^{2} x\).

Confirm that rewriting expressions using fundamental identities like the Pythagorean identity helps in matching equivalent expressions and completing the identity.

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Key Concepts

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

Pythagorean Identities

Pythagorean identities relate the squares of sine and cosine functions, such as sin²x + cos²x = 1. These identities are fundamental for rewriting expressions like cos²x in terms of sine or other trigonometric functions.

Trigonometric Expression Manipulation

Manipulating trigonometric expressions involves rewriting functions using identities or algebraic techniques to simplify or match given forms. This skill is essential when matching expressions from different columns or proving identities.

Understanding Function Notation and Powers

Recognizing that cos²x means (cos x)² is crucial for correctly applying identities and rewriting expressions. Misinterpreting notation can lead to errors in simplification or matching equivalent expressions.

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i & j Notation

Related Practice

Textbook Question

Verify that each equation is an identity.

(sec α + csc α) (cos α - sin α) = cot α - tan α

Textbook Question

The half-angle identity

tan A/2 = ± √[(1 - cosA)/(1 + cos A)]

can be used to find tan 22.5° = √(3 - 2√2), and the half-angle identity

tan A/2 = sin A/(1 + cos A)

can be used to find tan 22.5° = √2 - 1. Show that these answers are the same, without using a calculator. (Hint: If a > 0 and b > 0 and a² = b², then a = b.)

Textbook Question

Perform each transformation. See Example 2.

Write cot x in terms of csc x.

Textbook Question

Verify that each equation is an identity.

(sin² θ)/cos θ = sec θ - cos θ

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Textbook Question

Verify that each equation is an identity.

(csc θ + cot θ)/(tan θ + sin θ) = cot θ csc θ

Textbook Question

Let csc x = -3. Find all possible values of (sin x + cos x)/sec x.