Problem 12 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The identity is verified using the double angle formula: \( \cos^{2}(3x) - \sin^{2}(3x) = \cos(6x) \).

Step 1: Recall the Double Angle Identity

The identity we need to verify is \( \cos^{2}(3x) - \sin^{2}(3x) = \cos(6x) \). Recall the double angle formula for cosine: \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \). This formula closely resembles the left side of what we want to prove.

Step 2: Apply the Double Angle Identity to the Expression

Notice that the left side \( \cos^{2}(3x) - \sin^{2}(3x) \) matches the form of the double angle identity \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \), where \( \theta = 3x \). Therefore, we can directly apply this identity:\[\cos^{2}(3x) - \sin^{2}(3x) = \cos(2 \cdot 3x) = \cos(6x)\]

Step 3: Conclude the Verification

Since both sides of the equation \( \cos^{2}(3x) - \sin^{2}(3x) \) and \( \cos(6x) \) are now shown to be equal through the application of the double angle identity, the original statement is verified as an identity.

Key Concepts

Double Angle Formulas

In trigonometry, double angle formulas are incredibly useful tools. These formulas relate trigonometric functions of double angles (such as \(2\theta\)) to single angles (\(\theta\)). This makes them highly applicable in various mathematical problems, including simplification, integration, and identity verification.
The most common double angle formulas are:

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

These formulas can transform expressions with squared terms into simpler trigonometric terms or even to other functions. In the problem at hand, the expression \( \cos^2(3x) - \sin^2(3x) \) aligns perfectly with the double angle formula for cosine, leading to \( \cos(6x) \), confirming our use of this powerful tool. It's crucial to be familiar with these formulas as they often help solve and verify trigonometric identities quickly and efficiently.

Trigonometric Functions

Trigonometric functions form the foundation of trigonometry and are essential in describing the relationships between angles and sides in right triangles. There are six basic trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent.
Here's a quick overview:

• **Sine (sin)**: It represents the ratio of the opposite side to the hypotenuse in a right triangle.
• **Cosine (cos)**: It shows the ratio of the adjacent side to the hypotenuse.
• **Tangent (tan)**: It is the ratio of the opposite side to the adjacent side.

The other three—cosecant, secant, and cotangent—are the reciprocals of sine, cosine, and tangent respectively.
In the context of our exercise, understanding cosine is particularly vital. The formula \( \cos^2(\theta) - \sin^2(\theta) = \cos(2\theta) \) exploits the properties of these trigonometric functions to express the relationship between double angles clearly. Such knowledge allows mathematicians and students alike to tackle more complex equations by breaking them down into simpler components.

Verifying Identities

Verifying trigonometric identities is a fundamental skill in trigonometry. It involves proving that two expressions involving trigonometric functions are equivalent. This is done through algebraic manipulation and using known identities.
Key strategies include:

• Substituting known identities into the expression.
• Simplifying expressions step by step.
• Converting all functions to one kind for easier comparison, such as transforming everything into sines and cosines.
• Factorization or combining fractions where applicable.

In our exercise, we verify the identity \( \cos^2(3x) - \sin^2(3x) = \cos(6x) \) by recognizing the double angle formula for cosine. This allows us to show that the left side can transform directly into the right side, confirming their equivalence.
The goal when verifying identities is to show step-by-step that both sides of the equation are identical under certain conditions. Building proficiency in this area enhances problem-solving skills and deepens understanding of trigonometric concepts.