/*! 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 24 Prove that each of the following... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 24

Prove that each of the following identities is true.\((1-\cos x)(1+\cos x)=\sin ^{2} x\)

Short Answer

Expert verified
The identity is true: \((1-\cos x)(1+\cos x)=\sin^2 x\).

Step by step solution

01

Review the Given Identity

We need to prove that the identity \((1 - \cos x)(1 + \cos x) = \sin^2 x\) is true. This involves showing that both sides of the equation are equivalent for any value of \(x\).
02

Expand the Left Side

Begin by expanding the left side of the identity, \((1 - \cos x)(1 + \cos x)\), using the difference of squares formula. This gives us \(1^2 - (\cos x)^2 = 1 - \cos^2 x\).
03

Use the Pythagorean Identity

Recall the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). Solving for \(\sin^2 x\), we get \(\sin^2 x = 1 - \cos^2 x\).
04

Substitute Pythagorean Identity into Expanded Expression

Substitute \(1 - \cos^2 x\) from Step 2 with \(\sin^2 x\) from Step 3. So now, the left side \(1 - \cos^2 x\) becomes \(\sin^2 x\), proving that the identity is true.

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.

Pythagorean Identity
The Pythagorean Identity is a fundamental concept in trigonometry that relates the sine and cosine of an angle. It states that \[ \sin^2 x + \cos^2 x = 1 \].This equation expresses the relationship between the two primary trigonometric functions, showing that the square of the sine of any angle plus the square of the cosine of the same angle always equals one. This identity is derived from the Pythagorean Theorem and is useful in proving other trigonometric identities.
  • Solving for \( \sin^2 x \) gives \( \sin^2 x = 1 - \cos^2 x \).
  • Solving for \( \cos^2 x \) gives \( \cos^2 x = 1 - \sin^2 x \).
The Pythagorean Identity is often used to simplify expressions or to transform expressions in order to prove or verify trigonometric identities.
Difference of Squares
The difference of squares formula is a handy algebraic tool that allows us to simplify expressions of the form \((a - b)(a + b)\) into a more manageable form: \[ a^2 - b^2 \]. This formula states that the product of a conjugate pair (one minus and one plus) is equal to the square of the first element minus the square of the second. In the context of trigonometric expressions,
  • The difference of squares allows us to simplify \( (1 - \cos x)(1 + \cos x) \) into \( 1^2 - (\cos x)^2 = 1 - \cos^2 x \).
This expansion is crucial in the proof of certain identities. It often makes it possible to employ other identities, like the Pythagorean Identity, to further simplify the expression or complete a proof.
Trigonometric Proofs
Trigonometric proofs involve demonstrating that an equation involving trigonometric functions is true for all permissible values of the variable. These proofs can be aided by using established identities such as the Pythagorean Identity. For instance, in proving that \[ (1 - \cos x)(1 + \cos x) = \sin^2 x \], major steps include
  • Expanding \( (1 - \cos x)(1 + \cos x) \) using the difference of squares to get \( 1 - \cos^2 x \).
  • Applying the Pythagorean Identity to substitute \( 1 - \cos^2 x \) with \( \sin^2 x \).
Through these methods, you maintain mathematical equivalence as you manipulate the equation. By using these identities and proving methods, you can show consistency in trigonometric relationships and strengthen your ability to solve complex trigonometric problems.

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

Let \(\tan \theta=3\) with \(\theta\) in QI and find the following. $$ \cos 2 \theta $$

Rewrite each expression as a product. Simplify if possible. \(\sin 75^{\circ}-\sin 15^{\circ}\)

Prove each of the following identities. $$ \csc \theta-2 \sin \theta=\frac{\cos 2 \theta}{\sin \theta} $$

Let \(\csc t=\sqrt{5}\) with \(t\) in QII and find the following. $$ \cos 2 t $$

Graph each of the following from \(x=0\) to \(x=2 \pi\). $$ y=4 \cos ^{2} x-2 $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

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.