/*! 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 69 59鈥�76 Prove the identity. $$... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 69

59鈥�76 Prove the identity. $$\cos ^{4} x-\sin ^{4} x=\cos 2 x$$

Short Answer

Expert verified
\(\cos^4 x - \sin^4 x = \cos 2x\) by using the difference of squares and double angle identity.

Step by step solution

01

Recognize the Formula

The expression given is a difference of squares: \(\cos^4 x - \sin^4 x\). Utilize the identity for the difference of squares, \(a^2 - b^2 = (a + b)(a - b)\).
02

Apply the Difference of Squares Formula

Apply the formula to rewrite \(\cos^4 x - \sin^4 x\) as \((\cos^2 x + \sin^2 x)(\cos^2 x - \sin^2 x)\).
03

Simplify the Expression

Recall the Pythagorean identity \(\cos^2 x + \sin^2 x = 1\). Substitute this identity into the expression to simplify it to \(1(\cos^2 x - \sin^2 x)\).
04

Use Double Angle Identity

Recognize that \(\cos^2 x - \sin^2 x\) is equivalent to the double angle identity for cosine, \(\cos 2x\).
05

Write the Final Simplified Expression

Thus, the expression \(\cos^4 x - \sin^4 x\) simplifies to \(\cos 2x\), proving the identity.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of squares
The difference of squares is a fundamental algebraic concept frequently used to simplify expressions. It refers to an equation of the form \(a^2 - b^2\), which can be factored into \((a + b)(a - b)\). This makes solving expressions more straightforward by breaking them into simpler parts.
  • In the expression \(\cos^4 x - \sin^4 x\), we see it follows the pattern of a difference of squares.
  • Here, \(a\) is \(\cos^2 x\) and \(b\) is \(\sin^2 x\).
So, applying the difference of squares, \(\cos^4 x - \sin^4 x\) becomes \((\cos^2 x + \sin^2 x)(\cos^2 x - \sin^2 x)\). This manipulation simplifies our original equation into a more manageable form, setting the stage for further simplifications.
Pythagorean identity
The Pythagorean identity is a key trigonometric identity that states \(\cos^2 x + \sin^2 x = 1\). This simple yet powerful relationship helps us manage equations involving trigonometric functions. Here's how it assists in our problem:
  • After the first application of the difference of squares, we have the expression \((\cos^2 x + \sin^2 x)(\cos^2 x - \sin^2 x)\).
  • Using the Pythagorean identity, the term \(\cos^2 x + \sin^2 x\) simplifies to 1.
This reduction further simplifies our equation to \(1(\cos^2 x - \sin^2 x)\), effectively leaving us with just \(\cos^2 x - \sin^2 x\). Our equation becomes easier to handle and sets the stage for applying additional identities.
Double angle identity
The double angle identity for cosine is a trigonometric formula that expresses \(\cos 2x\) in terms of \(\cos x\) and \(\sin x\). One version of this identity states that \(\cos 2x = \cos^2 x - \sin^2 x\).
  • In the previous step, we derived the expression \(\cos^2 x - \sin^2 x\).
  • This matches perfectly with the double angle identity for cosine.
By recognizing this, the expression \(\cos^2 x - \sin^2 x\) can now be rewritten simply as \(\cos 2x\). This transformation confirms the original equation \(\cos^4 x - \sin^4 x = \cos 2x\). The use of trigonometric identities, like the double angle identity, allows us to verify and simplify expressions involving trigonometric functions effectively.

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

Verify the identity. $$ \frac{1}{\sec x+\tan x}+\frac{1}{\sec x-\tan x}=2 \sec x $$

Verify the identity. $$ (\sin x+\cos x)^{2}=1+2 \sin x \cos x $$

Verify the identity. $$ \sec ^{4} x-\tan ^{4} x=\sec ^{2} x+\tan ^{2} x $$

Verify the identity. $$ \frac{\cos x}{\sec x}+\frac{\sin x}{\csc x}=1 $$

Verify the identity. $$ (\sin x+\cos x)^{4}=(1+2 \sin x \cos x)^{2} $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

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.