Problem 56 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$\cos ^{2} 2 \theta-\sin ^{2} 2 \theta=2 \cos ^{2} 2 \theta-1$$

Step-by-Step Solution

Answer

The identity is verified: both expressions equal \(\cos 4\theta\).

Step 1: Identify the Identity

We are given the expression \(\cos ^{2} 2 \theta - \sin ^{2} 2 \theta\) and need to show it is equal to another expression \(2 \cos ^{2} 2 \theta - 1\). This resembles the double angle trigonometric identity for cosine: \(\cos 2x = \cos^2 x - \sin^2 x = 2 \cos^2 x - 1\).

Step 2: Apply the Cosine Double Angle Identity

The left-hand-side \(\cos^2 2\theta - \sin^2 2\theta\) can be recognized as the cosine double angle identity, which states \( \cos 2x = \cos^2 x - \sin^2 x \). Therefore, this expression simplifies to \(\cos 4\theta\).

Step 3: Re-express using Cosine Double Angle Identity

To link \(\cos 4\theta\) with the right-hand side, utilize the double angle identity in the form \(\cos 2x = 2 \cos^2 x - 1\). By substituting, we also see that \( \cos 4\theta = 2 \cos^2 2\theta - 1 \).

Step 4: Conclude the Identity Verification

Since \(\cos^2 2\theta - \sin^2 2\theta = \cos 4\theta\), and \(\cos 4\theta\) is also expressed as \(2 \cos^2 2\theta - 1\), the transformation confirms both sides are equal. Thus, the identity holds true.

Key Concepts

Cosine Double Angle Identity

The cosine double angle identity is a key formula in trigonometry that helps in simplifying expressions involving angles. This identity is typically written in three forms, showing the relationship between the cosine of an angle and the functions of its double:

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

These three versions are derived from the Pythagorean identity \( \cos^2 x + \sin^2 x = 1 \), which allows transformations and rearrangements.
Using the cosine double angle identities enables us to express trigonometric functions of double angles in terms of squares of trigonometric functions of a single angle, facilitating easier computations and simplification processes. In our exercise, the task utilized the identity \( \cos 2x = \cos^2 x - \sin^2 x \) to compare and recognize the equivalence of two expressions.

Trigonometric Transformations

A key part of verifying trigonometric identities is using transformations to simplify or rewrite expressions. In the context of the cosine double angle, we're often transforming a complex expression into a simpler, equivalent one.
The transformation process involves recognizing the forms of known identities, such as the cosine double angle identity, and applying these identities to make expressions more manageable. For example, seeing \( \cos^2 2\theta - \sin^2 2\theta \) and identifying that it matches \( \cos 2x = \cos^2 x - \sin^2 x \) allows us to re-express the formula as a single trigonometric function like \( \cos 4\theta \).
This method highlights the power of understanding and recognizing core identities in trigonometry, which can turn potentially complicated calculations into more navigable tasks.

Trigonometric Equations

Solving trigonometric equations involves finding the angles that satisfy the equation. These equations can be intricate, often requiring transformations like the ones we've discussed to find solutions.
Trigonometric equations frequently involve identities to express complicated terms in simpler forms. In our case, equations involving double angles can be reduced using double angle identities to form equations with single trigonometric functions of simpler arguments.

• Through these transformations, we can easily compare and equate different expressions.
• This approach enables us to verify identities, as was done when confirming \( \cos^2 2\theta - \sin^2 2\theta = 2 \cos^2 2\theta - 1 \).

The goal is to manipulate both sides of an equation until they are clearly identical, achieving balance and proving the identity.