Problem 12 - Precalculus: Functions and Graphs

Question

Verify the identity. \(\cos ^{2} 3 x-\sin ^{2} 3 x=\cos 6 x\)

Step-by-Step Solution

Answer

The identity \( \cos^2 3x - \sin^2 3x = \cos 6x \) is verified using the double angle identity for cosine.

Step 1: Identify the Trigonometric Identity

The given identity is \( \cos^2 3x - \sin^2 3x = \cos 6x \). You may recognize that this resembles the double angle identity for cosine, \( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \). Here, \( \theta \) is \( 3x \).

Step 2: Apply Double Angle Identity for Cosine

According to the double angle identity, \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \). By substituting \( 3x \) for \( \theta \), you convert it to \( \cos(2 \cdot 3x) = \cos^2 3x - \sin^2 3x \), which simplifies to \( \cos 6x = \cos^2 3x - \sin^2 3x \).

Step 3: Verify Both Sides are Equivalent

The right-hand side of the equation in the identity is \( \cos 6x \), and the expression on the left-hand side simplifies to \( \cos 6x \) using the double angle identity. Therefore, both sides of the identity are equivalent, verifying the identity.

Key Concepts

Double Angle Formulas

Double angle formulas are key tools in trigonometry that allow us to express trigonometric functions of double angles in terms of functions of single angles. These formulas are particularly useful because they simplify complex expressions and make solving trigonometric equations more manageable.

• The double angle formula for cosine is: \( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \).
• This formula can also be rewritten using the Pythagorean identity, \( \cos^2 \theta + \sin^2 \theta = 1 \), which gives us two additional forms: \( \cos(2\theta) = 2\cos^2 \theta - 1 \) and \( \cos(2\theta) = 1 - 2\sin^2 \theta \).

Replace \( \theta \) with any angle, such as \( 3x \), to handle specific identities, as demonstrated in the original exercise where \( \cos^2 3x - \sin^2 3x = \cos 6x \).
Understanding how to manipulate these identities is crucial for verifying equations and identities in trigonometry.

Cosine Function

The cosine function is one of the primary trigonometric functions, often paired with the sine function. Its graph is a wave that repeats every \( 2\pi \) radians.

• Cosine is an even function, meaning \( \cos(-x) = \cos(x) \).
• It starts at 1 when \( x = 0 \) and oscillates between -1 and 1.

In the context of double angle identities, the cosine function can be expressed in terms of other trigonometric functions like sine for angles such as \( \theta \) or \( 2\theta \). This capability is powerful for verifying trigonometric identities and solving trigonometric equations.
Graphical understanding of the cosine function helps in visualizing how the identities work and why these simplifications make sense mathematically.

Verifying Identities

Verifying trigonometric identities involves showing that two different expressions are equivalent for all angles where both sides are defined. This usually requires transforming one side of the equation to match the other.

• Identify which identity or formula is relevant to the problem. This could be a Pythagorean identity, a sum-to-product identity, or, as in the example, a double angle identity.
• Use algebraic manipulation to transform one side of the equation using known identities.
• Check that after simplification, both sides of the identity are indeed equal.

Verification can often involve a series of substitutions and simplifications. It’s like a puzzle that requires logic and practice to solve efficiently.
The given example of \( \cos^2 3x - \sin^2 3x = \cos 6x \) is directly verified by recognizing and applying the correct double angle identity, showing it holds true by equating both sides.