Problem 61 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The equation is an identity because both sides simplify to the same expression.

Step 1: Understand the Problem

We need to verify whether the equation \( \sec^2 \frac{x}{2} = \frac{2}{1+\cos x} \) holds true for all values of \( x \) within the domain of the expression. An identity is an equation that is true for all values of the variable for which both sides are defined.

Step 2: Express the Left Side in Terms of Cosine

The left side of the equation is \( \sec^2 \frac{x}{2} \), which by definition equals \( \frac{1}{\cos^2 \frac{x}{2}} \). Hence, \[ \sec^2 \frac{x}{2} = \frac{1}{\cos^2 \frac{x}{2}}. \]

Step 3: Simplify the Right Side

The right side of the equation is \( \frac{2}{1+\cos x} \). No simplification is directly evident yet, so let's work to express both sides using trigonometric identities.

Step 4: Use a Trigonometric Identity (Half Angle Identity)

Using the cosine half-angle identity, \( \cos \frac{x}{2} = \sqrt{\frac{1+\cos x}{2}} \), we know that \( \cos^2 \frac{x}{2} = \left(\sqrt{\frac{1+\cos x}{2}}\right)^2 = \frac{1+\cos x}{2} \).

Step 5: Equate Both Sides

Substitute \( \cos^2 \frac{x}{2} = \frac{1+\cos x}{2} \) into the expression for \( \sec^2 \frac{x}{2} \): \[ \sec^2 \frac{x}{2} = \frac{1}{\cos^2 \frac{x}{2}} = \frac{1}{\frac{1+\cos x}{2}} = \frac{2}{1+\cos x}. \]Since both sides simplify to the same expression, the equation is verified as an identity.

Key Concepts

Understanding the Secant Function

The secant function, denoted as \( \sec(x) \), is a trigonometric function that is the reciprocal of the cosine function. In mathematical terms, it is defined as:

• \( \sec(x) = \frac{1}{\cos(x)} \)

Understanding this relationship is crucial because it helps us transition between the two functions easily. From the definition, we see that the secant function is only defined when the cosine function is non-zero.
Because the cosine function can take values from -1 to 1, secant has vertical asymptotes where the cosine becomes zero. This happens at odd multiples of \( \frac{\pi}{2} \), such as \( \frac{\pi}{2}, \frac{3\pi}{2}, \ldots \)
In the context of the given exercise, the expression \( \sec^2 \frac{x}{2} \) involves using this reciprocal relationship, squared. Thus, \( \sec^2 \frac{x}{2} \) is equivalent to \( \frac{1}{\cos^2 \frac{x}{2}} \), indicating the importance of understanding secant deeply when dealing with trigonometric identities.

Exploring the Cosine Function

The cosine function is one of the primary trigonometric functions and is fundamental in the study of trigonometry. It is often described and graphed over its period, which is \( 2\pi \).

• \( \cos(x) \) represents the x-coordinate of a point on the unit circle.
• The function varies smoothly from -1 to 1 and back.

In the unit circle, the cosine of an angle x corresponds to the horizontal distance from the origin to a point on the circle.
This function returns a value of one at \( x = 0 \), and zero at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).
The role of cosine is pivotal in the given problem. We're concerned with \( \cos\frac{x}{2} \), specifically its square, as it directly ties into rewriting the secant function. Understanding how cosine behaves helps us comprehend the transition between angles and their impact on the identities we verify.

Half-Angle Identities

Half-angle identities are formulas that express trigonometric functions of half angles \( \frac{x}{2} \) in terms of functions of the original angle \( x \). They are incredibly useful in simplifying expressions and solving equations.

• \( \cos \frac{x}{2} = \sqrt{\frac{1+\cos x}{2}} \)
• \( \sin \frac{x}{2} = \sqrt{\frac{1-\cos x}{2}} \)

These identities make it possible to transform an expression involving a half angle into a form that involves the full angle. For example, knowing \( \cos^2 \frac{x}{2} \) turns into \( \frac{1+\cos x}{2} \) provides the key step in verifying the identity in our exercise.
The importance of half-angle identities is significantly highlighted when breaking down \( \sec^2 \frac{x}{2} \), allowing adjustment to match equivalent expressions of an equation using broader trigonometric identities. Thus, these identities are an essential tool in the toolkit for anyone exploring advanced trigonometry problems.