Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The identity \( \cos^4 x - \sin^4 x = \cos 2x \) is verified.

Step 1: Rewrite the left-hand side using the difference of squares

The expression \( \cos^4 x - \sin^4 x \) is a difference of squares. We can rewrite it as \( (\cos^2 x)^2 - (\sin^2 x)^2 \). This matches the form \( a^2 - b^2 = (a-b)(a+b) \).

Step 2: Factor the difference of squares

Apply the difference of squares formula: \( \cos^4 x - \sin^4 x = (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x) \).

Step 3: Simplify \( \cos^2 x + \sin^2 x \)

Since \( \cos^2 x + \sin^2 x = 1 \) (Pythagorean identity), the expression becomes \( (\cos^2 x - \sin^2 x)(1) = \cos^2 x - \sin^2 x \).

Step 4: Recognize \( \cos^2 x - \sin^2 x \) as a trigonometric identity

The expression \( \cos^2 x - \sin^2 x \) is a known trigonometric identity for \( \cos 2x \). Therefore, \( \cos^4 x - \sin^4 x = \cos 2x \).

Step 5: Conclude the identity verification

We've shown that \( \cos^4 x - \sin^4 x = \cos 2x \) holds true, verifying the identity as correct.

Key Concepts

Difference of Squares

In mathematics, a difference of squares is a particular case of polynomial identity that simplifies products of terms. Specifically, the expression has the form \( a^2 - b^2 \), which can be rewritten as \( (a-b)(a+b) \). This is incredibly useful in algebra for simplifying expressions and solving equations more efficiently.

In the given exercise, \( \cos^4 x - \sin^4 x \) resembles this identity. By setting \( a = \cos^2 x \) and \( b = \sin^2 x \), you can transform \( \cos^4 x - \sin^4 x \) into \( (\cos^2 x)^2 - (\sin^2 x)^2 \), fitting perfectly into the \( a^2 - b^2 \) mold.

Applying the difference of squares formula, it simplifies to:

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

This breakdown is essential as it assists in further simplification steps and solving the identity.

Pythagorean Identity

The Pythagorean Identity in trigonometry is a fundamental relation between sine and cosine functions. It states that for any angle \( x \), \( \cos^2 x + \sin^2 x = 1 \). This identity stems from the Pythagorean theorem and is valid for all real numbers.

In the exercise, after applying the difference of squares, we encounter the term \( \cos^2 x + \sin^2 x \). According to the Pythagorean Identity, this simplifies to 1.

By substituting 1 into the expression, we get:

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

This simplifies the problem significantly and is a key step in verifying the original identity.

Trigonometric Functions

Trigonometric functions are the core foundation of trigonometry and are integral to defining relationships in right-angled triangles. The primary trigonometric functions are sine, cosine, and tangent. They help describe the angle, size, and position of geometrical figures.

For this problem specifically, recognizing \( \cos^2 x - \sin^2 x \) as a known trigonometric identity is crucial. This equals \( \cos 2x \), which is the double angle identity expression.

This revelation leads directly to the equation from the problem:

• \( \cos^4 x - \sin^4 x = \cos 2x \)

Understanding these functions and their interrelated identities not only verifies the given problem but also deepens your comprehension of how sine and cosine interact within an identity framework. These tools are powerful when tackling more complex mathematical problems.