Problem 14 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\cos ^{4} 2 \theta+\sin ^{2} 2 \theta=\cos ^{2} 2 \theta+\sin ^{4} 2 \theta$$

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to equivalent expressions.

Step 1: Express Trigonometric Identities

We will start by using a couple of well-known trigonometric identities: \( \cos^2 x + \sin^2 x = 1 \) and the power-reduction identity, \( \cos^4 x = (\cos^2 x)^2 \) and \( \sin^4 x = (\sin^2 x)^2 \). These identities will be helpful for simplifying the expression.

Step 2: Simplify LHS

The left-hand side (LHS) of the equation is \( \cos^4 2\theta + \sin^2 2\theta \). Using the identity, \( \cos^4 2\theta = (\cos^2 2\theta)^2 \) and simplifying gives us \( \cos^4 2\theta = \cos^2 2\theta \cdot \cos^2 2\theta \). We also know that from \( \sin^2 2\theta \), we can express it in terms of cosine as \( 1 - \cos^2 2\theta \).

Step 3: Simplify RHS

The right-hand side (RHS) is \( \cos^2 2\theta + \sin^4 2\theta \). Using the identity for \( \sin^4 2\theta \), we know it's equal to \( (\sin^2 2\theta)^2 \). If we expand this, \( \sin^4 2\theta = \sin^2 2\theta \cdot \sin^2 2\theta \), or \( (1 - \cos^2 2\theta)^2 \).

Step 4: Equate Simplified Expressions

To prove the identity, we equate the simplified LHS, \( (\cos^2 2\theta)^2 + (1 - \cos^2 2\theta) \), with the simplified RHS, \( \cos^2 2\theta + (1 - \cos^2 2\theta)^2 \). Both simplify to the same identity upon further simplification, checking the consistency.

Key Concepts

Power-Reduction Identity

The power-reduction identity is a fundamental concept in trigonometry. It is used to express higher powers of trigonometric functions in terms of first powers. This can significantly simplify computations and verify complex identities.

For example, the power-reduction of \(\cos^4 x \) into \((\cos^2 x)^2 \) is very helpful. Similarly, \(\sin^4 x \) can be written as \((\sin^2 x)^2 \).

These identities play a crucial role when working with trigonometric functions:

• \( \cos^4 x = (\cos^2 x)^2 \)
• \( \sin^4 x = (\sin^2 x)^2 \)

Using these, along with other basic identities like \( \cos^2 x + \sin^2 x = 1 \), allows us to transform complex expressions for solving and proving identities.

Trigonometric Simplification

Simplifying trigonometric expressions means breaking them down using identities to make them easier to manage or compare. This often involves rewriting complex terms or changing forms using known trigonometric identities such as the Pythagorean identity.

To simplify the left-hand side (LHS) of the equation \( \cos^4 2\theta + \sin^2 2\theta \), we used the identity \( \cos^4 2\theta = (\cos^2 2\theta)^2 \) and expressed \( \sin^2 2\theta \) as \( 1 - \cos^2 2\theta \).

By applying simplification:

• Transform complex powers to simpler expressions
• Use Pythagorean identity to relate sine and cosine terms

The goal is to have both sides of the identity looking similar, allowing verification.

Trigonometric Functions

The core of trigonometry lies in understanding the trigonometric functions: sine, cosine, tangent, and their reciprocals. These functions relate angles in triangles to the ratios of a triangle's sides and are fundamental in many areas of math and science.

When verifying identities, knowing these functions and their relationships through identities is crucial. For instance, in our exercise, understanding that \( \cos^2 x + \sin^2 x = 1 \) is instrumental. This identity helps in expressing one function in terms of others.
In applying these:

• Angles are often doubled or halved (e.g., \( 2\theta \))
• Powers of trigonometric functions often need reducing for simplification

Proper manipulation and understanding of trigonometric functions allow for efficient verification and simplification in proofs.