Problem 74 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The equation is an identity because both sides simplify to \( \cos x \).

Step 1: Recognize the Identity

The equation we are trying to verify as an identity is \( \frac{\sin 2x}{2 \sin x} = \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2} \). The right-hand side, \( \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2} \), can be rewritten using the double-angle identity for cosine, which is \( \cos^2 \theta - \sin^2 \theta = \cos 2\theta \). Therefore, the right-hand side simplifies to \( \cos x \).

Step 2: Simplify the Left-Hand Side

Now let's simplify the left-hand side of the equation: \( \frac{\sin 2x}{2 \sin x} \). We know from the double angle formula that \( \sin 2x = 2 \sin x \cos x \). So we replace \( \sin 2x \) with \( 2 \sin x \cos x \) in the left-hand side:\[\frac{2 \sin x \cos x}{2 \sin x} \]

Step 3: Cancel Common Terms

In the expression \( \frac{2 \sin x \cos x}{2 \sin x} \), the numerators and denominators both contain \( 2 \sin x \). Therefore, we can cancel \( 2 \sin x \) from the numerator and denominator, leaving us with:\[\cos x \]

Step 4: Compare Both Sides

After simplification, the left-hand side becomes \( \cos x \). Earlier, we simplified the right-hand side also to \( \cos x \). Since both sides are equal, we have verified that \( \frac{\sin 2x}{2 \sin x} = \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2} \) is an identity.

Key Concepts

Double Angle Formulas

The double angle formulas are powerful tools in trigonometry used to express trigonometric functions of double angles, such as \( \sin 2x \) or \( \cos 2x \), in terms of single angles \( x \). These formulas can greatly simplify expressions and computations in trigonometry.

Consider the formula for \( \sin 2x \):

• \( \sin 2x = 2 \sin x \cos x \)

This means the sine of a double angle is twice the product of the sine and cosine of the single angle.

Similarly, the formula for \( \cos 2x \) can be given in three different forms:

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

These equations help us rewrite expressions in various useful ways depending on the context of the problem. Being familiar with these formulas is crucial for solving trigonometrical equations efficiently.

Simplifying Expressions

Simplifying expressions in trigonometry involves using identities and algebraic skills to make expressions easier to handle or compare.

In the problem provided, we simplified the expression \( \frac{\sin 2x}{2 \sin x} \) using the double angle formula \( \sin 2x = 2 \sin x \cos x \). Notice how the expression here was simplified by:

• Replacing \( \sin 2x \) with \( 2 \sin x \cos x \).
• Cancelling the common term \( 2 \sin x \) in the numerator and denominator.

The result was the much simpler expression \( \cos x \), which made it easier to compare with the other side of the equation.

When simplifying, always look for common trigonometric identities or factors you can cancel. Reduction leads to simpler, more manageable forms, facilitating further calculations or verifications.

Verifying Identities

Verifying identities is a key skill in trigonometry, which allows us to demonstrate that two trigonometric expressions are equal for all values in their domain. This process often involves both analytical manipulation and logical reasoning.

Let's break down how you can verify an identity like the one in the exercise:

• **Recognize familiar patterns**: Look for ways to use known identities, such as the double angle formulas.
• **Simplify expressions**: Use trigonometric identities and algebra to simplify the expressions on both sides of the equation.
• **Compare both sides**: Once simplified, check if both expressions are indeed equal.

In our example, simplifying both sides of the equation led to \( \cos x \).

By showing that the simplified form of both sides is identical, we successfully verified the identity. Practice and familiarity with various trigonometric identities and properties are key to mastering this concept. As you work through problems, keep a list of useful identities at hand for quicker reference.