Problem 22 - Precalculus: Functions and Graphs

Question

Verify the identity. \(\cos ^{4} x-\sin ^{4} x=\cos 2 x\)

Step-by-Step Solution

Answer

Identity verified: \( \cos^4 x - \sin^4 x = \cos 2x \).

Step 1: Recognize the Difference of Squares

The expression \( \cos^4 x - \sin^4 x \) is a difference of squares, which we can factor using the identity \( a^2 - b^2 = (a-b)(a+b) \). In our case, \( a = \cos^2 x \) and \( b = \sin^2 x \).

Step 2: Factor the Expression

Applying the difference of squares identity, we get:\[ \cos^4 x - \sin^4 x = (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x) \].

Step 3: Simplify Using Pythagorean Identity

From the Pythagorean identity, we know \( \cos^2 x + \sin^2 x = 1 \). Substitute this in to simplify:\[ (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x) = (\cos^2 x - \sin^2 x)(1) = \cos^2 x - \sin^2 x \].

Step 4: Recognize the Cosine Double Angle Identity

The expression \( \cos^2 x - \sin^2 x \) is equivalent to \( \cos 2x \), according to the cosine double angle identity.

Step 5: Verify the Identity

Thus, we have shown that \( \cos^4 x - \sin^4 x = \cos 2x \), verifying the given identity.

Key Concepts

Difference of Squares

In mathematics, the difference of squares is an important concept and appears frequently in expressions that need simplification or factoration. The difference of squares formula states that for any two expressions, say \( a \) and \( b \), the formula can be written as:

• \( a^2 - b^2 = (a-b)(a+b) \)

This formula is very useful for breaking down more complex expressions into simpler factors, which can be easier to manage.
In the original problem, \( \cos^4 x - \sin^4 x \) aligns with the difference of squares format. Here, \( \cos^2 x \) acts as \( a \), and \( \sin^2 x \) as \( b \). Thus, the expression can be factored as:

• \( (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x) \)

Recognizing when to use this identity is essential in simplifying trigonometric equations.

Pythagorean Identities

The Pythagorean identities are fundamental relationships in trigonometry that are derived from the Pythagorean Theorem. They help in simplifying trigonometric expressions and confirming identities.

• One basic Pythagorean identity is \( \cos^2 x + \sin^2 x = 1 \)

This identity states that in a right-angled triangle, the sum of the squares of the sine and cosine of any angle \( x \) is always equal to 1.
In the original exercise, during the simplification step, this identity makes the factor \( \cos^2 x + \sin^2 x \) reduce to 1:

• Thus, changing \( (\cos^2 x - \sin^2 x)(1) \) into \( \cos^2 x - \sin^2 x \)

Using Pythagorean identities is crucial to reduce the expressions to simpler forms, thus making complex equations manageable.

Cosine Double Angle Identity

The cosine double angle identity is another vital tool when working with trigonometric identities. It offers a way to express the cosine of a double angle in terms of the squares of sine and cosine of a single angle.

• The identity is given by: \( \cos 2x = \cos^2 x - \sin^2 x \)

This identity helps in rephrasing equations to verify identities or find values of trigonometric functions.
In this exercise, the expression \( \cos^2 x - \sin^2 x \) exactly matches the cosine double angle identity. Thus, it is equivalent to \( \cos 2x \).

• Using this identity confirms that \( \cos^4 x - \sin^4 x = \cos 2x \)

Understanding and applying the cosine double angle identity can simplify solving complex trigonometric equations and prove helpful when transforming expressions.