Problem 67 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

Yes, the given equation is an identity as both sides equal \( \frac{1 - \cos x}{2} \).

Step 1: Understand the Identity

The equation given is supposed to be an identity, implying it should hold true for all valid values of \( x \). The equation is \( \sin^2 \frac{x}{2} = \frac{\tan x - \sin x}{2 \tan x} \). Our goal is to verify that the left side is identically equal to the right side.

Step 2: Use Trigonometric Identities

Start with the left side of the equation: \( \sin^2 \frac{x}{2} \). Use the identity \( \sin^2 \theta = \frac{1 - \cos 2\theta}{2} \). So, \( \sin^2 \frac{x}{2} = \frac{1 - \cos x}{2} \).

Step 3: Simplify the Right Side

The right side of the equation is \( \frac{\tan x - \sin x}{2 \tan x} \). Replacing \( \tan x \) with \( \frac{\sin x}{\cos x} \), this becomes:\( \frac{\frac{\sin x}{\cos x} - \sin x}{2 \cdot \frac{\sin x}{\cos x}} \).

Step 4: Simplify the Expression Further

Simplify by finding a common denominator: \( \frac{\sin x - \sin x \cos x}{\cos x} \). The numerator simplifies to \( \sin x (1 - \cos x) \). The entire expression then becomes:\( \frac{\sin x (1 - \cos x)/\cos x}{2 \sin x / \cos x} = \frac{1 - \cos x}{2} \) when reducing the term.

Step 5: Compare Both Sides

Now, the right side simplifies to \( \frac{1 - \cos x}{2} \). Substitute the left side: \( \sin^2 \frac{x}{2} = \frac{1 - \cos x}{2} \). Since both sides of the equation are equal, the identity is verified.

Key Concepts

Verify an Identity

Verifying a trigonometric identity is a process of proving that the given equation is true for all values of the variable involved, as long as the expressions are defined. When you encounter an identity to verify, such as \( \sin^2 \frac{x}{2} = \frac{\tan x - \sin x}{2 \tan x} \), it means that both sides of the equation are universally equivalent.

To verify an identity, you typically need to manipulate one or both sides of the equation using known trigonometric identities and algebraic operations until they match perfectly. Here’s a step-by-step approach you can follow:

• Understand the given identity and the trigonometric functions involved.
• Choose a side of the identity to work on. Simplifying the more complex side first is often helpful.
• Apply appropriate trigonometric identities and simplifications to reach the same form as the other side.
• Check for logical algebraic transformations, keeping the range of valid \( x \) values in mind.

By practicing these methods, you become adept at recognizing and verifying many types of trigonometric identities.

Trigonometric Simplification

Trigonometric simplification involves reducing expressions into simpler or more manageable forms while conserving the equivalency of the original equation. This is essential in solving equations, verifying identities, and making complex expressions more comprehensible.

For instance, consider the expression \( \frac{\tan x - \sin x}{2 \tan x} \). By rewriting \( \tan x \) as \( \frac{\sin x}{\cos x} \), the expression becomes \( \frac{\frac{\sin x}{\cos x} - \sin x}{2 \cdot \frac{\sin x}{\cos x}} \). Here are several simplification techniques:

• Rewrite functions using basic identities, like replacing \( \tan x \) with \( \frac{\sin x}{\cos x} \).
• Look for common denominators to simplify fractions.
• Factor expressions when possible, such as taking out common terms in a numerator or denominator.
• Cancel, combine, or reorganize terms to achieve a form that is easier to handle or compare.

Trigonometric simplification requires familiarity with various identities, such as Pythagorean, angle sum/difference, and double angle identities, aiding in reshaping the expressions effectively.

Half-Angle Identities

Half-angle identities are a subset of trigonometric identities that simplify expressions involving angles that are divided in half, such as \( \sin^2 \frac{x}{2} \). These identities help in rewriting trigonometric functions to reveal relations that can make solving equations easier.

The half-angle identity for sine is particularly useful:

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

Here’s how it works in context:
You start with \( \sin^2 \frac{x}{2} \) and apply the half-angle identity, transforming it into \( \frac{1 - \cos x}{2} \). This expression is much simpler and aligns with commonly used trigonometric forms.

Half-angle identities are instrumental in connecting single angle functions to their half-angles, assisting in solving trigonometric equations, especially those involving transformations, integrations, or verifications in angle manipulations. Mastering these identities, makes trigonometry a lot more intuitive and manageable.