Problem 76 - Precalculus Mathematics for Calculus

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; both sides simplify to \(\frac{\cos x+\sin x}{\cos x-\sin x}\).

Step 1: Verify the Identity Structure

We need to show that the expression \(\frac{1+\tan x}{1-\tan x}\) on the left is equivalent to \(\frac{\cos x + \sin x}{\cos x - \sin x}\) on the right. First, recognize the structure of both sides to understand what transformations might be needed.

Step 2: Rewrite \(\tan x\) in Terms of Sine and Cosine

Express \(\tan x\) as \(\frac{\sin x}{\cos x}\). This changes the left side to: \[\frac{1+\frac{\sin x}{\cos x}}{1-\frac{\sin x}{\cos x}}.\] Simplifying gives \[ \frac{\cos x + \sin x}{\cos x - \sin x} \], which matches the right-hand side.

Step 3: Simplifying the Left Side Fraction

To simplify \(\frac{1+\frac{\sin x}{\cos x}}{1-\frac{\sin x}{\cos x}}\), multiply both the numerator and the denominator by \(\cos x\):\[\frac{\cos x(1+\frac{\sin x}{\cos x})}{\cos x(1-\frac{\sin x}{\cos x})}\] simplifies to \[\frac{\cos x + \sin x}{\cos x - \sin x}.\]

Step 4: Confirm Both Sides are Equal

After simplifying the left side expression, we have \(\frac{\cos x + \sin x}{\cos x - \sin x}\). It is identical to the right-hand side expression that was given initially. Hence, the identity is verified.

Key Concepts

Tangent Function

The tangent function, noted as \( \tan x \), is one of the primary functions in trigonometry, alongside sine and cosine. It represents the ratio of the opposite to the adjacent side in a right triangle. Mathematically, it can be expressed as:

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

This relationship, expressing it in terms of sine and cosine, becomes particularly useful when working with trigonometric identities. By rewriting the tangent in terms of sine and cosine, it becomes easier to simplify expressions or verify identities.The tangent function is periodic, with a period of \( \pi \). It also has vertical asymptotes where \( \cos x = 0 \), which occur at odd multiples of \( \frac{\pi}{2} \). This is important to note because at these points, the function is undefined. Understanding these properties of the tangent function can simplify the process of solving trigonometric equations and identities.

Sine and Cosine Relationship

The sine and cosine functions are fundamental components of trigonometry. They are related through Pythagorean identities and ratios helping solve various trigonometric problems. The basic identity between them is:

• \( \sin^2 x + \cos^2 x = 1 \)

This signifies that for any angle \( x \), the square of the sine of \( x \) plus the square of the cosine of \( x \) equals one, a vital identity to memorize.Sine and cosine are also the building blocks for other trigonometric functions including tangent. These functions are both periodic with a period of \( 2\pi \).Understanding these relationships can help in transforming expressions, like rewriting \( \tan x \) in terms of \( \sin x \) and \( \cos x \) or simplifying complex fractions as seen in identity verifications. These transformations are critical in tackling problems involving complex trigonometric identities.

Identity Verification

Verifying trigonometric identities involves showing that two different expressions, which may appear distinct, are actually equivalent for all values in their domain. This stems from fundamental identities and relationships between the trigonometric functions.To verify an identity, such as \( \frac{1+\tan x}{1-\tan x} = \frac{\cos x + \sin x}{\cos x - \sin x} \), you can follow steps to express all components in terms of sine and cosine. By substituting \( \tan x = \frac{\sin x}{\cos x} \), this identity transforms into a form that aligns with known expressions.The process:

• Rewrite all terms using basic trigonometric identities (e.g., express tangent as sine over cosine).
• Simplify expressions step-by-step, ensuring each transformation is valid.
• Perform algebraic manipulations to ensure both sides of the equation match clearly.

Complete and careful verification of identities requires a thorough understanding of trigonometric properties and an eye for simplification. Solutions can seem complex initially, but breaking them down into smaller, digestible steps can make them much easier to handle.