Problem 76 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \( \frac{\cos x + \sin x}{\cos x - \sin x} \).

Step 1: Express Tangents in Terms of Sine and Cosine

Recall that \( \tan x = \frac{\sin x}{\cos x} \). Replace \( \tan x \) in the left-side expression: \[ \frac{1+\frac{\sin x}{\cos x}}{1-\frac{\sin x}{\cos x}} \].

Step 2: Simplify the Expression Using Fraction Operations

Combine the terms in the numerator and denominator: \[ \frac{\frac{\cos x + \sin x}{\cos x}}{\frac{\cos x - \sin x}{\cos x}} \]. This simplifies to \( \frac{\cos x + \sin x}{\cos x - \sin x} \) after canceling out the \( \cos x \) in both the numerator and denominator.

Step 3: Compare Simplified Expressions

Both the left side \( \frac{\cos x + \sin x}{\cos x - \sin x} \) and the given right side \( \frac{\cos x + \sin x}{\cos x - \sin x} \) are identical, confirming that both expressions are equivalent.

Key Concepts

Tangent Function

The tangent function, often written as \( \tan x \), is a key trigonometric function. It relates the angle in a right triangle to the ratio of the opposite side to the adjacent side. But, it's more useful in identities and equations to use its relationship with sine and cosine.

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

This equation expresses tangent in terms of the sine and cosine functions, two more fundamental trigonometric functions. Understanding this is crucial when working with trigonometric identities or when simplifying expressions. In the given exercise, expressing the tangent in terms of sine and cosine was the first key step to verifying the identity. This transformation allows us to work with simpler fractions and apply other trigonometric properties to simplify the whole expression.

Sine and Cosine

Sine and cosine are the foundational blocks of trigonometry. They represent the fundamental trigonometric relationships in the unit circle.

• Sine \( (\sin x) \): measures the vertical position (or y-coordinate) on the unit circle.
• Cosine \( (\cos x) \): measures the horizontal position (or x-coordinate) on the unit circle.

Their relationship is pivotal when converting expressions and solving equations. In trigonometric identities, they can often simplify complexities by converting less common functions like tangent, secant, or cotangent into terms of sine and cosine. When the original exercise expressed \( \tan x \) as \( \frac{\sin x}{\cos x} \), it took advantage of this core relationship to reframe the problem in easier terms and simplify the final comparison between two expressions.

Fraction Simplification

Fraction simplification is a process of rewriting fractions in their simplest form, making them easier to understand or compare. It involves removing common factors from the numerator and the denominator.Here's how it's applied in the exercise:

• Combine fractions with the same denominators: This step might involve adding or subtracting numerators.
• Cancel common factors: If the numerator and denominator share a factor, it can be canceled out from the fraction.

In the exercise, the expression \( \frac{1+\frac{\sin x}{\cos x}}{1-\frac{\sin x}{\cos x}} \) was simplified by recognizing that both parts of the fraction have a common denominator of \( \cos x \). By combining terms to have a single denominator, it becomes clear how to cancel \( \cos x \) from both parts. This simplification ultimately leads to reaching the same expression on both sides of the equation as given in the original problem: \( \frac{\cos x + \sin x}{\cos x - \sin x} \). Simplifying fractions in this way is essential to verifying identities or solving equations successfully.