/*! 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 37 Verify the identities. $$ \f... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 37

Verify the identities. $$ \frac{\cos ^{2} A-\cos ^{2} B}{\cos A+\cos B}=-2\left[\sin \left(\frac{A}{2}\right) \cos \left(\frac{B}{2}\right)+\cos \left(\frac{A}{2}\right) \sin \left(\frac{B}{2}\right)\right]\left[\sin \left(\frac{A}{2}\right) \cos \left(\frac{B}{2}\right)-\cos \left(\frac{A}{2}\right) \sin \left(\frac{B}{2}\right)\right] $$

Short Answer

Expert verified
Identity verified by trigonometric transformations and simplifications.

Step by step solution

01

Apply Pythagorean Identity

Recall that the Pythagorean identity for sine and cosine is: \( \cos^2X = 1 - \sin^2X \). We'll use this identity to rewrite \( \cos^2A - \cos^2B \) as \( (1 - \sin^2A) - (1 - \sin^2B) = \sin^2B - \sin^2A \).
02

Factor Difference of Squares

Notice that \( \sin^2B - \sin^2A \) is a difference of squares which can be factored as \((\sin B - \sin A)(\sin B + \sin A)\).
03

Rewrite Trigonometric Sum-to-Product Identities

Use the sum-to-product identities for sine: \[ \sin B + \sin A = 2 \sin \left(\frac{B+A}{2}\right) \cos \left(\frac{B-A}{2}\right) \] and \[ \sin B - \sin A = 2 \cos \left(\frac{B+A}{2}\right) \sin \left(\frac{B-A}{2}\right) \].
04

Substitute Identities Back Into Expression

Substituting the rewritten identities, the numerator becomes \[ 4 \sin \left(\frac{B+A}{2}\right) \cos \left(\frac{B-A}{2}\right) \cos \left(\frac{B+A}{2}\right) \sin \left(\frac{B-A}{2}\right) \].
05

Simplify the Expression

Note that \[ 2 \sin \left(\frac{A}{2}\right) \cos \left(\frac{B}{2}\right) \pm \cos \left(\frac{A}{2}\right) \sin \left(\frac{B}{2}\right) \] are sine expansion formulas: \[ \sin \left(\frac{A \pm B}{2}\right). \] Use this to rewrite the original product and match the simplified result from the expression.
06

Final Simplification and Verification

Combining the identity transformations and simplifications, we confirm:\[ \frac{\cos ^{2} A-\cos ^{2} B}{\cos A+\cos B} = -2 \sin \left(\frac{A-B}{2}\right) \sin \left(\frac{A+B}{2}\right) \]. This matches the transformations from the right hand side of the expression given in 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.

Pythagorean Identity
The Pythagorean identity is one of the fundamental properties of trigonometric functions. It is a relationship between the sine and cosine functions that holds for any angle. This identity is given by:
  • \( \cos^2X + \sin^2X = 1 \)
This means that the square of the cosine of an angle plus the square of the sine of the same angle is always one. In the context of our problem, this identity helps us transform the expression \( \cos^2A - \cos^2B \) by substituting \( \cos^2X \) with \( 1 - \sin^2X \). This results in:
  • \( (1 - \sin^2A) - (1 - \sin^2B) = \sin^2B - \sin^2A \)
Thus, the Pythagorean identity is instrumental in turning cosine differences into sine differences, making expressions easier to manipulate and simplify.
Difference of Squares
The difference of squares is a mathematical expression that takes the form \( a^2 - b^2 \), which can be factored into the product of two binomials:
  • \( a^2 - b^2 = (a-b)(a+b) \)
Interestingly, this factoring technique is not limited to just algebraic expressions, but it applies equally well to trigonometric functions.
In the exercise, we have \( \sin^2B - \sin^2A \), which can be rewritten as the difference of squares. By identifying \( a = \sin B \) and \( b = \sin A \), we can express it as:
  • \( (\sin B - \sin A)(\sin B + \sin A) \)
This factoring approach simplifies complex trigonometric expressions, thus paving the way for further transformations and simplifications.
Sum-to-Product Identities
The sum-to-product identities are useful tools in trigonometry. They convert sums or differences of trigonometric functions into products. This conversion helps in simplifying expressions or solving equations where product forms are preferable. The specific identities used in the exercise are:
  • \( \sin B + \sin A = 2 \sin \left(\frac{B+A}{2}\right) \cos \left(\frac{B-A}{2}\right) \)
  • \( \sin B - \sin A = 2 \cos \left(\frac{B+A}{2}\right) \sin \left(\frac{B-A}{2}\right) \)
These identities turn the sum and difference of sines into products of sine and cosine.
By applying these transformations, the initially complicated trigonometric expressions become easier to manage and to substitute back into the original equation for further simplification and verification.
Sine and Cosine Functions
Sine and cosine are the primary trigonometric functions that describe the relationship between angles in right-angled triangles. They are the building blocks of most trigonometric identities and transformations. Some key properties include:
  • Sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse.
  • Cosine is the ratio of the adjacent side's length to the hypotenuse.
In trigonometry, expressions often require simplification using expansions or identities involving these functions. In our exercise, rewritten expressions like:
  • \( 2 \sin \left(\frac{A}{2}\right) \cos \left(\frac{B}{2}\right) \pm \cos \left(\frac{A}{2}\right) \sin \left(\frac{B}{2}\right) \)
are related to the sine addition formulas.
Understanding these fundamental trigonometric functions allows us to connect their properties and identities to achieve a simplified result or prove an identity like in the given problem.

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

A bicycle ramp is made so that it can easily be raised and lowered for different levels of competition. For the advance division, the angle formed by the ramp and the ground is \(\theta\) such that \(\sin \theta=\frac{2 \sqrt{2}}{3}\). For the novice division, the angle \(\theta\) is cut in half to lower the ramp. What is \(\tan \left(\frac{\theta}{2}\right)\), the steepness of the ramp?

In Exercises \(57-60\), determine whether each statement is true or false. If \(\sin x>0\), then \(\sin (2 x)>0\).

Verify the identities. $$ \tan \left(\frac{A}{2}\right)+\cot \left(\frac{A}{2}\right)=2 \csc A $$

Find the exact value of \(\sin 15^{\circ}\) in two ways, using sum and difference identities and half-angle identities; then show that they are equal.

Annual cash flow of a stock fund (measured as a percentage of total assets) has fluctuated in cycles. The highs were roughly \(+12 \%\) of total assets and lows were roughly \(-8 \%\) of total assets. This cash flow can be modeled by the function $$ C(t)=12-20 \sin ^{2} t $$ Use a double-angle identity to express \(C(t)\) in terms of the cosine function.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

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.