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Chapter 5: Problem 117

Verify the identity. $$\cos 4 \alpha=\cos ^{2} 2 \alpha-\sin ^{2} 2 \alpha$$

Short Answer

Expert verified
Using the double-angle formulas and Pythagorean identity, it was shown that \(\cos 4 \alpha=\cos^{2} 2 \alpha-\sin^{2} 2 \alpha \).

Step by step solution

01

Convert \(\cos 4\alpha\) into double-angle form

Using the double-angle formula, where \(\cos 2u = 2\cos^{2}u - 1\), we can write \(\cos 4\alpha\) as \(\cos 2(2\alpha)\) = \(2\cos^{2}(2\alpha) - 1\).
02

Rotate the Position of the Terms

We move the equation around to match the given equation. Thus, we have \( \cos 2(2\alpha)\) = \(\cos^{2}(2\alpha) - 1 +1\), which simplifies to \(\cos^{2}(2\alpha) - [1 - \cos 2(2\alpha)]\)
03

Apply the Pythagorean Identity

We can replace \(1 - \cos 2(2\alpha)\) with \(\sin^{2}2\alpha\), due to Pythagorean Identity where \(1 - \cos^{2}u = \sin^{2}u\). Thus, we obtain \( \cos^{2}2\alpha - \sin^{2}2\alpha\), which is the same as the equation in the exercise.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Double-Angle Formula
The double-angle formula is a powerful tool in trigonometry used to express trigonometric functions of double angles in terms of functions of single angles. In this context, it helps us simplify complex trigonometric expressions and solve identities. Here is what you need to know about the double-angle formula for cosine:
  • The formula states: \( \cos 2\theta = 2\cos^{2}\theta - 1 \).
  • This formula allows us to express \( \cos 4\alpha \) as \( \cos 2(2\alpha) \), which is \( 2\cos^{2}(2\alpha) - 1 \).
  • It is derived by using the sum identities for cosine since \( \cos(\theta + \theta) \) can be expressed as \( \cos^{2}\theta - \sin^{2}\theta \).
When using the double-angle formula, it becomes easier to transform identities and analyze their components effectively. This is the first step we take when verifying trigonometric identities like the one in the original exercise.
Pythagorean Identity
The Pythagorean identity plays a crucial role in trigonometry and is fundamental to establishing relationships between sine, cosine, and their complements. The identity is stated as:
  • \( \sin^2\theta + \cos^2\theta = 1 \).
This identity can be manipulated to replace terms depending on the expression in use:
  • For example, \( 1 - \cos^2\theta = \sin^2\theta \) and conversely, \( 1 - \sin^2\theta = \cos^2\theta \).
In the given identity verification, we used the Pythagorean identity to transform \( 1 - \cos 2(2\alpha) \) into \( \sin^2(2\alpha) \). This replacement is the key step in reaching the desired expression \( \cos^2(2\alpha) - \sin^2(2\alpha) \). Recognizing how to apply these relationships will greatly ease the process of verifying other trigonometric identities.
Verification of Identities
Verifying trigonometric identities involves showing that two sides of an equation represent the same expression, often by transforming one side to match the other using algebraic and trigonometric properties. Here鈥檚 a step-by-step approach for identity verification:
  • Step 1: Start with more complex or transforms-prone side, which often needs simplification.
  • Step 2: Utilize known identities such as double-angle or Pythagorean identities to rewrite expressions.
  • Standby step: Simplify fractions, factor common terms, and combine like terms as necessary.
  • Final step: Transform the equation to reveal transparent relationships or equality.
In our exercise, the task was to verify if \( \cos 4\alpha = \cos^2(2\alpha) - \sin^2(2\alpha) \). By breaking down the expressions using the double-angle and Pythagorean identities, we demonstrated the equivalence, completing the verification process. With practice, this step-by-step approach aids quick and efficient identity verification.

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