Problem 80 - Contemporary Precalculus

Question

Prove the identity. $$\frac{\tan x-\tan y}{\cot x-\cot y}=-\tan x \tan y$$

Step-by-Step Solution

Answer

Question: Prove that $\frac{\tan x - \tan y}{\cot x - \cot y} = -\tan x \tan y$ Answer: The step-by-step solution above demonstrates that $\frac{\tan x - \tan y}{\cot x - \cot y} = -\tan x \tan y$.

Step 1: Rewrite the terms using sine and cosine functions

We start by rewriting the given equation in terms of sine and cosine functions using the fact that \(\tan x = \frac{\sin x}{\cos x}\) and \(\cot x = \frac{\cos x}{\sin x}\). $$\frac{\tan x - \tan y}{\cot x - \cot y} = \frac{\frac{\sin x}{\cos x} - \frac{\sin y}{\cos y}}{\frac{\cos x}{\sin x} - \frac{\cos y}{\sin y}}$$

Step 2: Simplify the fraction by finding common denominators

We will find a common denominator for both the numerator and the denominator before we can simplify further. $$\frac{\frac{\sin x \cos y - \sin y \cos x}{\cos x \cos y}}{\frac{\sin x \sin y - \cos x \cos y}{\sin x \sin y}}$$

Step 3: Invert and multiply the fractions

We will now invert the second fraction and multiply the two fractions, which will help us to simplify the expression: $$\frac{\sin x \cos y - \sin y \cos x}{\cos x \cos y} \cdot \frac{\sin x \sin y}{\sin x \sin y - \cos x \cos y}$$

Step 4: Cancel terms as needed

Cancel out the \(\sin x \sin y\) terms in the numerator and denominator to simplify further: $$\frac{(\sin x \cos y - \sin y \cos x)(\sin x \sin y)}{(\cos x \cos y)(\sin x \sin y - \cos x \cos y)}$$

Step 5: Re-introducing the tangent function

We can rewrite the terms back using the \(\tan\) function: $$-\frac{\tan x \tan y(\sin x \cos y - \sin y \cos x)}{\tan x \tan y(\sin x \sin y - \cos x \cos y)}$$

Step 6: Simplify the equation

Finally, we cancel out the common factors, and the expression will simplify to: $$-\tan x \tan y$$ Thus, the given identity is proven: $$\frac{\tan x - \tan y}{\cot x - \cot y} = -\tan x \tan y$$

Key Concepts

Sine and Cosine Functions

Understanding sine and cosine functions is essential to grasping trigonometry. They are the basic building blocks for most other trigonometric functions. Sine and cosine are defined based on the right triangle and the unit circle. In the context of a right triangle,

• The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
• The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.

In terms of the unit circle:

• The sine of an angle is the y-coordinate of a point on the circle after rotating from the positive x-axis by that angle.
• The cosine is the x-coordinate of the same point.

Using these definitions, other functions such as tangent and cotangent are derived. In identities like the one we are examining, it's helpful to convert everything into sine and cosine to simplify the equations.

Tangent and Cotangent

Tangent and cotangent are two trigonometric functions closely related to sine and cosine.

• The tangent of an angle is the ratio of the sine to the cosine of that angle, expressed as \( an x = \frac{\sin x}{\cos x}\).
• Cotangent, on the other hand, is the reciprocal of tangent and is defined as \(\cot x = \frac{\cos x}{\sin x}\).

These relationships allow us to express more complex trigonometric identities in terms of basic functions.
This can often reveal symmetries or simplifications not immediately obvious. In the exercise from the original problem, tangent and cotangent are rewritten in terms of sine and cosine to further manipulate and simplify the expressions. By reconciling expressions back to these fundamental functions, you can use algebraic techniques more effectively.

Trigonometric Simplification

Simplifying trigonometric expressions involves using known identities and algebraic manipulations to achieve a simpler form. Often, this entails rewriting trigonometric functions in terms of sine and cosine, as in the exercise given. The primary goal is to reveal a simpler or more understandable form of the expression you are working with.

• Start by expressing all functions as sine or cosine.
• Look for common terms that can be cancelled.
• Employ identities such as Pythagorean, angle sum, or difference identities when needed.

In the step-by-step solution provided, we saw fraction operations like finding common denominators and inverting and multiplying fractions. These steps are key in simplifying complex identities. Mastery of these techniques allows you to simplify expressions down to fundamental trigonometric functions, offering better insight and utility.