Problem 14 - Precalculus with Limits

Question

In Exercises 9-50, verify the identity \( \cos^2 \beta - \sin^2 \beta = 2 \cos^2 \beta - 1 \)

Step-by-Step Solution

Answer

After substitution of \( \sin^2 \beta \) with \( 1 - \cos^2 \beta \) and simplifying, the left side becomes \( 2 \cos^2 \beta - 1 \), which verifies the original identity.

Step 1: Start with the Left Side

Begin by working with the left side of the original equation which is \( \cos^2 \beta - \sin^2 \beta \)

Step 2: Use Pythagorean Identity

Using the Pythagorean Identity \( \sin^2 \beta = 1 - \cos^2 \beta \), replace \( \sin^2 \beta \) in the left side of the equation. This will give you \( \cos^2 \beta - (1 - \cos^2 \beta) \)

Step 3: Simplify the expression

Simplify the expression to get \( 2 \cos^2 \beta - 1 \)

Step 4: Match Right Side

This result matches the right side of the original equation, thus confirming the identity \( \cos^2 \beta - \sin^2 \beta = 2 \cos^2 \beta - 1 \)

Key Concepts

Pythagorean identity

A cornerstone in trigonometry is the Pythagorean identity, which is essential in solving and verifying many trigonometric equations. The identity is derived from the Pythagorean theorem and can be written as:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity is fundamental because it allows you to express one trigonometric function in terms of another. For example, from \( \sin^2 \theta + \cos^2 \theta = 1 \), you can derive:

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

These formulas are incredibly useful when simplifying expressions, as seen in this exercise. By replacing \( \sin^2 \beta \) with \( 1 - \cos^2 \beta \), the angle reduction becomes more manageable, leading to the verification of the trigonometric identity presented.

Trigonometric equations

Trigonometric equations involve the relationship between angles and their trigonometric functions. To solve these, we often rely on known identities and algebraic manipulations. This simplifies complex expressions and reveals underlying properties. For example, in the exercise:

• We start with the equation \( \cos^2 \beta - \sin^2 \beta \)
• Apply the Pythagorean identity to transform the expression

Trigonometric equations may seem complex at first, but by systematically applying identities, we can solve or verify them. Often, the challenge is recognizing which identity to use and how to reconfigure the equation. Here, recognizing the Pythagorean identity allowed us to change the expression into a simpler form that mirrors the provided equation.

Angle simplification

Angle simplification involves reducing expressions or equations involving angles to a more straightforward form. In trigonometry, this simplification can come from applying identities like the Pythagorean identity, double angle formulas, or other trigonometric relationships. For simplifying \( \cos^2 \beta - \sin^2 \beta \) to \( 2 \cos^2 \beta - 1 \), we use:

• Rewrite \( \sin^2 \beta \) using the Pythagorean identity
• Simplify the resulting expression to make it identical to the other side of the equation

Angle simplification makes trigonometric expressions easier to analyze and solve. It is akin to solving a puzzle—replacing and rearranging pieces to see the whole picture. By mastering these techniques, you enhance your ability to work with complex trigonometric expressions effectively.