Problem 60 - A Graphical Approach to Precalculus with Limits

Question

Verify that each equation is an identity. $$\frac{\cos (A-B)}{\sin (A+B)}=\frac{1+\cot A \cot B}{\cot A+\cot B}$$

Step-by-Step Solution

Answer

The equation is an identity because both sides simplify to the same expression.

Step 1: Rewrite using Trigonometric Identities

To simplify the left side of the equation, use the identity for \( \cos(A-B) \): \[ \cos(A-B) = \cos A \cos B + \sin A \sin B \] and the identity for \( \sin(A+B) \):\[ \sin(A+B) = \sin A \cos B + \cos A \sin B \]Substituting these identities, our equation becomes:\[ \frac{\cos A \cos B + \sin A \sin B}{\sin A \cos B + \cos A \sin B} \]

Step 2: Express in terms of Cotangent

We know that \( \cot A = \frac{\cos A}{\sin A} \) and \( \cot B = \frac{\cos B}{\sin B} \). Rewrite the left side of the identity using these definitions:

Step 3: Simplify the Right Side

The right side of the equation, \( \frac{1 + \cot A \cot B}{\cot A + \cot B} \), can be rewritten using the definitions for cotangent:\[ \frac{1 + \frac{\cos A \cos B}{\sin A \sin B}}{\frac{\cos A}{\sin A} + \frac{\cos B}{\sin B}} = \frac{\frac{\sin A \sin B + \cos A \cos B}{\sin A \sin B}}{\frac{\cos A \sin B + \cos B \sin A}{\sin A \sin B}} \]

Step 4: Factor out the Common Denominator

The expression from Step 3 has a common denominator in both the numerator and the denominator, \( \sin A \sin B \). Cancel these from the entire fraction:\[ = \frac{\sin A \sin B + \cos A \cos B}{\cos A \sin B + \sin A \cos B} \]

Step 5: Compare Both Sides

Now notice that the simplified right side, \( \frac{\sin A \sin B + \cos A \cos B}{\cos A \sin B + \sin A \cos B} \), is equivalent to the left side, \( \frac{\cos A \cos B + \sin A \sin B}{\sin A \cos B + \cos A \sin B} \). Thus, both sides of the equation are equal, confirming it is an identity.

Key Concepts

Cosine of Sum and Difference

The formulas for the Cosine of Sum and Difference are fundamental tools in trigonometry. They help you break down more complex cosine angles into expressions that involve simpler trigonometric functions of individual angles. The identity for cosine difference, key in this exercise, is:

• \( \cos(A-B) = \cos A \cos B + \sin A \sin B \)

This formula shows you how to express the cosine of a difference between two angles as the sum of the products of cosines and sines of those angles.
When you deal with such identities, always try to rearrange and substitute the known values to simplify expressions.
It can turn complicated fractions into more workable forms using familiar trigonometric terms.

Sine of Sum and Difference

Similarly, there are identities for the Sine of Sum and Difference of two angles, another pivotal mathematical strategy.These identities allow you to convert a sine of a sum or difference into products of sine and cosine functions of individual angles. For the sum of angles, you have the identity:

• \( \sin(A+B) = \sin A \cos B + \cos A \sin B \)

Substituting sine sum identities simplifies trigonometric expressions just like it did in our exercise.
You start by replacing combined angle functions with expressions involving simple angle functions.This approach helps in reducing complex expressions into simpler, more familiar terms.
These identities are particularly useful when verifying trigonometric equations or identities.

Cotangent

Cotangent, abbreviated as \( \cot \), is another trigonometric function you often encounter alongside sine, cosine, and tangent.It is the reciprocal of the tangent function, defined as:

• \( \cot A = \frac{\cos A}{\sin A} \)

Understanding cotangent plays a crucial role in reworking trigonometric expressions.In this exercise, using the cotangent identities helps you break down the given complex fraction on the right side of the equation.Rewriting expressions involving cotangent can often lead to more simplified forms, as shown in the solution. This can help verify equations by providing alternative equivalent forms that may be easier to manipulate and compare with other expressions.

Simplification of Fractions

Fractional simplification is a critical math skill, especially essential in trigonometry.It involves reducing complex fractions into their simplest form, often by canceling out common factors.In this exercise, after substituting trigonometric identities, the next step was simplifying the fraction on the right side.This required recognizing and factoring out the common term \( \sin A \sin B \) in both the numerator and the denominator:

• \( \frac{\sin A \sin B + \cos A \cos B}{\cos A \sin B + \sin A \cos B} \)

Doing so helps you compare different expressions easily.
It assists in seeing that the two sides of the equation are identical, confirming the trigonometric identity effectively without complicated calculations.
Simplicity and clarity are the goals when working through such mathematical tasks.