Problem 62 - A Graphical Approach to Precalculus with Limits

Question

Verify that each equation is an identity. $$\sec 2 x=\frac{1+\tan ^{2} x}{1-\tan ^{2} x}$$

Step-by-Step Solution

Answer

The given equation is an identity because both sides simplify to \( \frac{1}{\cos 2x} \).

Step 1: Understand the Identity

We are given the equation \( \sec 2x = \frac{1 + \tan^2 x}{1 - \tan^2 x} \) and need to verify if it is true for all \( x \). An identity holds true for all values within the domain of the functions involved. First, we should recall that \( \sec 2x = \frac{1}{\cos 2x} \).

Step 2: Express in Terms of Sine and Cosine

Express both sides of the identity in terms of sine and cosine:- For the left side, \( \sec 2x = \frac{1}{\cos 2x} \).- For the right side, use the identity \( \tan x = \frac{\sin x}{\cos x} \), so \( \tan^2 x = \frac{\sin^2 x}{\cos^2 x} \). Substituting, we have:\[\frac{1 + \tan^2 x}{1 - \tan^2 x} = \frac{1 + \frac{\sin^2 x}{\cos^2 x}}{1 - \frac{\sin^2 x}{\cos^2 x}} = \frac{\frac{\cos^2 x + \sin^2 x}{\cos^2 x}}{\frac{\cos^2 x - \sin^2 x}{\cos^2 x}}\]

Step 3: Simplify the Right Hand Side

Notice that \( \cos^2 x + \sin^2 x = 1 \). Therefore, the right side simplifies to:\[\frac{1}{\frac{\cos^2 x - \sin^2 x}{\cos^2 x}} = \frac{\cos^2 x}{\cos^2 x - \sin^2 x} = \frac{\cos^2 x}{\cos 2x}\] since \( \cos 2x = \cos^2 x - \sin^2 x \). Then, the right side becomes: \( \frac{1}{\cos 2x} \).

Step 4: Conclude the Verification

Both sides of the equation have now been simplified to \( \frac{1}{\cos 2x} \), confirming that the given equation \( \sec 2x = \frac{1 + \tan^2 x}{1 - \tan^2 x} \) is indeed an identity. Thus, it holds true for all \( x \) where the expressions are defined.

Key Concepts

Secant Function

The secant function, denoted as \( \sec x \), is one of the six fundamental trigonometric functions. It is the reciprocal of the cosine function, meaning that \( \sec x = \frac{1}{\cos x} \). The secant function is particularly interesting because it tends to exhibit vertical asymptotes, which occur at the zeros of the cosine function.

Aside from its relationship with the cosine function, the secant function finds its application in various scenarios, such as in calculating the magnitudes of vectors and in waveforms.

• When examining functions like \( \sec 2x \), we apply transformations to the standard secant function, which involves stretching the curve horizontally, making it useful in complex equations and identities.
• Understanding the secant function in relation to the cosine function helps in verifying and manipulating trigonometric identities effectively.

By deepening your grasp of the secant function, you're not just learning an equation but understanding a critical aspect of trigonometry that connects several other components.

Tangent Function

The tangent function, represented by \( \tan x \), is another fundamental trigonometric function. It is defined as the ratio of the sine and cosine functions, expressed as \( \tan x = \frac{\sin x}{\cos x} \).

In practical terms, the tangent function describes the slope of the line formed by an angle in the unit circle and is essential in determining angles in various calculations like elevations and depressions.

• One of the key properties of the tangent function is that it is periodic with a period of \( \pi \), meaning it repeats its values in intervals of \( \pi \).
• Like the secant function, the tangent function also exhibits asymptotic behavior, specifically around points where its denominator, \( \cos x \), equals zero.

The tangent function's usefulness extends well beyond its simplicity, serving as the basis for developing critical trigonometric identities and solving complex angular relationships.

Pythagorean Identity

The Pythagorean identity is one of the most important identities in trigonometry, establishing a fundamental link among the sine, cosine, and secant functions. It is usually expressed as \( \sin^2 x + \cos^2 x = 1 \).

This relation stems from the unit circle, where any point on the circle can be represented as \( (\cos x, \sin x) \), leading to the equation of the circle being \( x^2 + y^2 = 1 \).

• The identity allows for transformations and simplifications of more complex expressions by substituting one part of the identity to simplify trigonometric equations.
• It also provides a pathway to derive other forms of the identity, such as \( 1 + \tan^2 x = \sec^2 x \), by dividing the initial identity by \( \cos^2 x \).

Utilizing the Pythagorean identity extensively in solving mathematical problems fosters a deeper understanding of the interconnectedness of trigonometric functions, and aids in comprehending their behaviors and transformations.