Problem 84 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\cot x+1}{\cot x-1}=\frac{1+\tan x}{1-\tan x} $$

Step-by-Step Solution

Answer

The identity is verified.

Step 1: Express Cotangent in Terms of Tangent

We know that \( \cot x = \frac{1}{\tan x} \). Substitute this expression into the left side of the identity: \( \frac{\cot x + 1}{\cot x - 1} = \frac{\frac{1}{\tan x} + 1}{\frac{1}{\tan x} - 1} \).

Step 2: Simplify the Left Side

To simplify, multiply the numerator and the denominator by \( \tan x \) to eliminate the fractions: \[\frac{(\frac{1}{\tan x} + 1) \times \tan x}{(\frac{1}{\tan x} - 1) \times \tan x} = \frac{1 + \tan x}{1 - \tan x}.\]

Step 3: Compare Both Sides of the Equation

After simplification, we have \( \frac{1 + \tan x}{1 - \tan x} = \frac{1 + \tan x}{1 - \tan x} \), which shows both sides are identical.

Key Concepts

Cotangent

Cotangent is one of the six fundamental trigonometric functions and is often represented as \( \cot x \). It is the reciprocal of the tangent function and can be defined as \( \cot x = \frac{1}{\tan x} \). In the context of a right triangle, cotangent represents the ratio of the adjacent side to the opposite side.
Understanding the relationship between cotangent and tangent is important when verifying trigonometric identities. In many problems, expressing trigonometric functions in terms of others can simplify complex expressions or identify patterns. For example, to simplify \( \frac{\cot x + 1}{\cot x - 1} \), knowing \( \cot x \) as \( \frac{1}{\tan x} \) is the key first step that allows for further simplification by eliminating complicated fractions.
In equations and identities, recognizing which functions are inverses or reciprocals can lead to significant simplifications. This process reveals that often, what seems complex can become straightforward through substitution and equivalent transformations.

Tangent

Tangent is an essential trigonometric function, usually denoted as \( \tan x \). It represents the ratio of the opposite side to the adjacent side in a right triangle. As one of the core functions in trigonometry, it frequently appears in various mathematical and engineering fields.
Identities and equations involving tangent often require transformation and substitution strategies for simplification. For example, when working with the identity \( \frac{1 + \tan x}{1 - \tan x} \), writing the expression of cotangent in terms of tangent can greatly simplify the process. Doing so aligns all terms under a uniform function, making manipulation and recognition of equal forms much easier.
Moreover, tangent's periodic and symmetric properties are often exploited to solve equations or verify identities. By understanding these characteristics and how tangent interacts with other trigonometric functions, approaching problems becomes more intuitive and manageable.

Simplification of Fractions

Simplifying fractions is a fundamental skill in algebra and trigonometry, often used in proving identities or solving equations. In this particular exercise, one essential technique is multiplying both the numerator and denominator of a fraction by a common factor to eliminate complex fractions. This is often done to ease comparison or calculation tasks.
For instance, when faced with \( \frac{\frac{1}{\tan x} + 1}{\frac{1}{\tan x} - 1} \), multiplying through by \( \tan x \) removes the fractions in each term, transforming the expression into \( \frac{1+\tan x}{1-\tan x} \).
This simplification is critical for directly comparing two algebraic expressions. It helps in revealing hidden equalities between complicated functions and reduces potential errors in manual calculations. Practicing these simplification methods enhances problem-solving skills, especially for more advanced trigonometric expressions.