Problem 84 - Precalculus Mathematics for Calculus

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 as both sides become \( \frac{1+\tan x}{1-\tan x} \).

Step 1: Understand the Identity

We need to verify the identity: \( \frac{\cot x + 1}{\cot x - 1} = \frac{1 + \tan x}{1 - \tan x} \). This identity asserts that two expressions are equivalent for all values of \( x \) where they are defined.

Step 2: Express in Terms of Tangent

Recall that \( \cot x = \frac{1}{\tan x} \). Substituting \( \cot x \) in terms of \( \tan x \), the left side becomes: \( \frac{\frac{1}{\tan x} + 1}{\frac{1}{\tan x} - 1} \).

Step 3: Simplify the Left Side

Simplify \( \frac{\frac{1}{\tan x} + 1}{\frac{1}{\tan x} - 1} \) by finding a common denominator for both the numerator and denominator: \( \frac{\frac{1 + \tan x}{\tan x}}{\frac{1 - \tan x}{\tan x}} \).

Step 4: Simplify Further

The complex fraction becomes \( \frac{1 + \tan x}{1 - \tan x} \) after canceling \( \tan x \) from both the numerator and denominator.

Step 5: Compare Both Sides

After simplification, the left side \( \frac{1 + \tan x}{1 - \tan x} \) matches with the right side. Thus, both sides of the identity are equal.

Key Concepts

Cotangent

Cotangent, often denoted as \( \cot x \), is a fundamental trigonometric function closely related to tangent. It is defined as the reciprocal of the tangent function:

• \( \cot x = \frac{1}{\tan x} \)
• \( \cot x = \frac{\cos x}{\sin x} \)

This means the cotangent function is the ratio of the adjacent side to the opposite side in a right triangle. In terms of unit circle, it's the horizontal distance divided by the vertical distance from the origin to a point on the unit circle. Cotangent plays a vital role in verifying and simplifying trigonometric identities.
It often appears when converting or simplifying equations involving both cosine and sine functions.

Tangent

Tangent, written as \( \tan x \), is another essential trigonometric function. It is the ratio of the sine of an angle to its cosine:

• \( \tan x = \frac{\sin x}{\cos x} \)

On a right triangle, tangent represents the ratio of the length of the opposite side to the adjacent side. For the unit circle, it's the vertical distance over the horizontal distance from the origin.
It is commonly used in all trigonometric identities and simplifications due to its direct relationship with sine and cosine.

The tangent function is crucial when converting complex trigonometric expressions since it often directly relates to the cotangent function as well.

Simplification of Expressions

Simplification of expressions is a fundamental skill in algebra and trigonometry. The goal is to reduce complex expressions to a simpler or more understandable form, while maintaining their equivalence. In our task, we simplified the left side of the given identity by:

• Substituting \( \cot x \) using \( \frac{1}{\tan x} \)
• Finding a common denominator for fractions
• Canceling common factors

These steps are crucial because they help you manipulate and adjust components of an expression to make comparisons or verify identities possible. Simplification allows you to see the underlying patterns or relationships between different trigonometric expressions, leading to accurate conclusions about their equivalence.

Verification of Identities

Verification of identities in trigonometry involves proving that two expressions are equivalent for all values in their domain. This process usually requires algebraic manipulations and substitutions. To verify the identity given in the example:

• Understand what the identity claims
• Express functions in terms of simpler expressions or related trigonometric identities
• Simplify step by step to show both sides are identical
• Compare the final forms of each side as evidence of equivalence

By thorough investigation and manipulation using known identities, we establish that both sides of the given equation are equal. This is a powerful method in trigonometry that assures the correctness and integrity of formulas used in calculations.