Problem 39 - Precalculus with Limits

Question

In Exercises 9-50, verify the identity \( \dfrac{\tan x + \cot y}{\tan x \cot y} = \tan y + \cot x \)

Step-by-Step Solution

Answer

Yes, the given trigonometric identity \( \dfrac{\tan x + \cot y}{\tan x \cot y} = \tan y + \cot x \) is indeed correct.

Step 1: Understand the trigonometric identities

It's important to know that \(\tan x = \dfrac{1}{\cot x}\) and \(\cot x = \dfrac{1}{\tan x}\). This will help you manipulate the equation to match both sides.

Step 2: Apply the identities

Apply the identities to the left-hand side of the equation. Rewrite \(\tan x\) as \(\dfrac{1}{\cot x}\) and \(\cot y\) as \(\dfrac{1}{\tan y}\). Now the equation becomes \(\dfrac{\dfrac{1}{\cot x} + \dfrac{1}{\tan y}}{\dfrac{1}{\cot x} * \dfrac{1}{\tan y}}\).

Step 3: Simplify the equation

Note that this is a complex fraction. Simplify it by multiplying the numerator and denominator by \(\tan y * \cot x\). The equation then transforms to \(\dfrac{\tan y + \cot x}{1}\).

Step 4: Verify the identity

Now the left-hand side of the equation matches with the right-hand side. Therefore, it verifies the given trigonometric identity.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable where both sides of the equation are defined. These identities are crucial in simplifying and solving trigonometry problems as they relate various trigonometric functions to one another.

For example, one of the most fundamental identities is the Pythagorean identity which relates the sine and cosine of an angle: \[ \sin^2x + \cos^2x = 1 \.\] By understanding these relationships, you can transform complex expressions into simpler forms, making them easier to work with. When working with trigonometric identities, be sure to familiarize yourself with the basic ones such as reciprocal identities, quotient identities, and Pythagorean identities. This foundational knowledge will enable you to move through verifying trigonometric identities with greater ease.

Tan and Cot Relationships

The tangent (\(\tan\)) and cotangent (\(\cot\)) functions are part of the six primary trigonometric functions, and they have a reciprocal relationship. This means that \[ \tan x = \frac{1}{\cot x} \] and \[ \cot x = \frac{1}{\tan x} \.\] These relationships reveal that knowing the value of either function allows you to easily find the value of the other.

Often, in trigonometry exercises, you'll come across expressions that involve both tangent and cotangent, and transforming one into the reciprocal of the other can simplify the equation drastically. This strategy is especially useful when verifying trigonometric identities or simplifying complex trigonometric expressions.

Simplifying Complex Fractions

Complex fractions are fractions where the numerator, the denominator, or both are also fractions. Simplifying them might initially seem daunting, but it's mainly about finding a common denominator and making use of the relationships between trigonometric functions.

When dealing with complex trigonometric fractions, multiplying the numerator and denominator by a common factor which is the product of all individual denominators in the complex fraction can eliminate the smaller fractions within. This method, known as the 'LCD method' (least common denominator), transforms the complex fraction into a simpler one which is often easier to solve or simplify. For trigonometric equations, this might mean multiplying by expressions such as \(\tan x \cot x\) or \(\sin x \csc x\), depending on the functions involved.

Once simplified, these fractions can be further reduced by applying trigonometric identities or arithmetic operations to arrive at a solution or verify an identity.