/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 Factor the expression and use th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 25

Factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer. $$1-2 \cos ^{2} x+\cos ^{4} x$$

Short Answer

Expert verified
The simplified form of the given expression is \(\sin^4x\)

Step by step solution

01

Recognize the Expression as a Quadratic

Firstly, note that the given expression \(1-2 \cos ^{2} x+\cos ^{4} x\) can be viewed as quadratic in form if we replace \(\cos^2 x\) by \(y\). The expression then resembles \(1-2y+y^2\), which can be factored easily.
02

Factor the Quadratic Expression

We proceed to factor the quadratic expression. In the place of \(y\), we substitute \(\cos^2 x\). With the use of the identity \(a^2 - 2ab + b^2 = (a-b)^2\), we may rewrite the expression as \((1-\cos^2x)^2\).
03

Use Fundamental Identity to Further Simplify

At this point, we can use the fundamental Pythagorean identity \(1-\cos^2x = \sin^2x\). Substituting this into our previously obtained result gives us the final solution of \((\sin^2x)^2\), or we can also write it as \(\sin^4x\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use the sum-to-product formulas to find the exact value of the expression. $$\sin 75^{\circ}+\sin 15^{\circ}$$

Use the sum-to-product formulas to find the exact value of the expression. $$\cos 120^{\circ}+\cos 60^{\circ}$$

Find all solutions of the equation in the interval \([0,2 \pi) .\) Use a graphing utility to graph the equation and verify the solutions. $$\frac{\cos 2 x}{\sin 3 x-\sin x}-1=0$$

Find the exact value of the expression. $$\frac{\tan (5 \pi / 6)-\tan (\pi / 6)}{1+\tan (5 \pi / 6) \tan (\pi / 6)}$$

Use the half-angle formulas to simplify the expression. $$-\sqrt{\frac{1-\cos 8 x}{1+\cos 8 x}}$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.