Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity. $$ \frac{\sin u-\sin v}{\cos u-\cos v}=-\cot \frac{1}{2}(u+v) $$

Step-by-Step Solution

Answer

The identity is verified as true by simplifying to \(-\cot \frac{1}{2}(u+v)\).

Step 1: Recall the Identities

To verify the trigonometric identity, recall some fundamental trigonometric identities. The difference of sines can be expressed using the identity: \( \sin u - \sin v = 2 \cos \frac{u+v}{2} \sin \frac{u-v}{2} \). Similarly, the difference of cosines can be expressed as: \( \cos u - \cos v = -2 \sin \frac{u+v}{2} \sin \frac{u-v}{2} \).

Step 2: Substitute the Identities

Substitute these identities into the given expression: \( \frac{\sin u - \sin v}{\cos u - \cos v} = \frac{2 \cos \frac{u+v}{2} \sin \frac{u-v}{2}}{-2 \sin \frac{u+v}{2} \sin \frac{u-v}{2}} \).

Step 3: Simplify the Expression

Cancel out the common terms \( \sin \frac{u-v}{2} \) in the numerator and the denominator: \( \frac{2 \cos \frac{u+v}{2}}{-2 \sin \frac{u+v}{2}} = -\frac{\cos \frac{u+v}{2}}{\sin \frac{u+v}{2}} \).

Step 4: Recognize the Cotangent

Recognize that the expression \(-\frac{\cos \frac{u+v}{2}}{\sin \frac{u+v}{2}}\) is \(-\cot \frac{1}{2}(u+v)\), which matches the identity to be verified.

Key Concepts

Difference of Sines

The difference of sines identity is an essential tool in trigonometry. It provides a way to express the difference between two sine values in terms of other trigonometric functions. The identity is given by: \[\sin u - \sin v = 2 \cos \frac{u+v}{2} \sin \frac{u-v}{2}\]This formula can be useful in various trigonometric transformations and verifications. Notice how it breaks down the difference of sines into a product involving the cosine and sine of the averages and half-differences of the angles.

• \(u\) and \(v\) are the angles in question.
• \(\cos\) and \(\sin\) signify trigonometric functions of cosine and sine.

Using this identity can simplify complex trigonometric expressions and is particularly helpful in proving other identities.

Difference of Cosines

The identity for the difference of cosines gives a similar advantage as the difference of sines. It allows us to transform a subtraction of cosines into a product involving sines. The identity is:\[\cos u - \cos v = -2 \sin \frac{u+v}{2} \sin \frac{u-v}{2}\]Here, notice that the result is a product of sines. This transformation is particularly powerful when simplifying expressions or verifying identities involving differences of cosines.

• The negative sign indicates a phase shift or direction change in the trigonometric function.
• The factors involve both the sum and the difference of the angles \(u\) and \(v\).

This identity is a cornerstone in manipulations and simplifications within trigonometry.

Cotangent

The cotangent function is one of the primary trigonometric functions. It relates to the tangent function and is defined as the reciprocal of the tangent:\[\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}\]In the context of our problem, recognizing the structure \(-\frac{\cos \frac{u+v}{2}}{\sin \frac{u+v}{2}}\) as the cotangent is key. This reflects the cotangent of the half-angle of \(u+v\):

• This recognition simplifies many trigonometric expressions.
• Cotangent helps in expressing angles in terms of ratios of sides in right-angled triangles.

Understanding the cotangent and its properties expands the toolkit for tackling a variety of trigonometric problems.

Simplifying Expressions

Simplifying trigonometric expressions is a fundamental skill. It involves using identities, like those for the difference of sines and cosines, to transform complex equations into simpler forms.The goal in simplification can vary:

• Making an expression easier to interpret or calculate.
• Verifying identities or proving certain properties.

For example, in our exercise, you initially have:\[\frac{\sin u - \sin v}{\cos u - \cos v}\]Substituting the differences of sines and cosines identities, then canceling and simplifying, we reached \[-\frac{\cos \frac{u+v}{2}}{\sin \frac{u+v}{2}} = -\cot \frac{1}{2}(u+v)\]Effective simplification often involves pattern recognition and a strong grasp of fundamental trigonometric identities.