Problem 42 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ \cos ^{4} w+1-\sin ^{4} w=2 \cos ^{2} w $$

Step-by-Step Solution

Answer

The identity \( \cos^4 w + 1 - \sin^4 w = 2 \cos^2 w \) holds true.

Step 1: Express Identity in Terms of Cosine and Sine

We start by expressing the equation \( \cos^4 w + 1 - \sin^4 w = 2 \cos^2 w \) in terms of basic trigonometric identities. Notice that \( \cos^4 w \) and \( \sin^4 w \) can be factored by using the identity \( \sin^2 w + \cos^2 w = 1 \). Therefore, we have:\( \cos^4 w = (\cos^2 w)^2 \) and \( \sin^4 w = (\sin^2 w)^2 \).

Step 2: Use Difference of Squares

Now, rewrite \( \cos^4 w - \sin^4 w \) as a difference of squares:\[(\cos^2 w)^2 - (\sin^2 w)^2 = (\cos^2 w - \sin^2 w)(\cos^2 w + \sin^2 w)\]Since \( \cos^2 w + \sin^2 w = 1 \), the expression simplifies to:\[\cos^2 w - \sin^2 w\]

Step 3: Simplify and Verify the Right Side

Substitute the simplified left side \( \cos^2 w - \sin^2 w \) back into the original expression:\[\cos^2 w - \sin^2 w + 1 = 2 \cos^2 w\]Isolate terms involving \( \cos^2 w \) by rearranging:\[\cos^2 w - \sin^2 w + 1 = \cos^2 w + \cos^2 w\]\[\cos^2 w - \sin^2 w = \cos^2 w\]Therefore, the identity \( \cos^4 w + 1 - \sin^4 w = 2 \cos^2 w \) is verified.

Key Concepts

Cosine and Sine Relationship

Understanding the relationship between cosine and sine is crucial in trigonometry. The most fundamental identity that connects these two is the Pythagorean identity:

• \( \sin^2 w + \cos^2 w = 1 \)

This identity is extremely useful because it allows us to express one trigonometric function in terms of the other. For example, if you need to find \( \sin^2 w \) and you know \( \cos^2 w \), you can simply rearrange the identity to:

• \( \sin^2 w = 1 - \cos^2 w \)

Similarly, transforming expressions like \( \cos^4 w \) can be easier when using their squared counterparts:

• \( \cos^4 w = (\cos^2 w)^2 \)

Being comfortable with the cosine and sine relationship enables you to manipulate and work through more complex trigonometric expressions with greater ease. This foundational relationship serves as a tool for verifying identities and simplifying expressions.

Difference of Squares

The difference of squares is a powerful algebraic tool used to simplify expressions. It is applied when you have an expression like \( a^2 - b^2 \), which can be factored into

• \((a - b)(a + b)\)

In the trigonometric identity \( \cos^4 w - \sin^4 w \), we used this principle as follows:

• \( (\cos^2 w)^2 - (\sin^2 w)^2 = (\cos^2 w - \sin^2 w)(\cos^2 w + \sin^2 w) \)

Remembering that \( \cos^2 w + \sin^2 w = 1 \), the expression simplifies to \( \cos^2 w - \sin^2 w \).
This technique of transforming an expression's structure using the difference of squares helps in simplifying complex trigonometric identities, so they become easier to solve or verify.

Verification of Trigonometric Identities

Verifying trigonometric identities is a process of proving that one side of an equation can be transformed into the other. The key lies in using known identities and algebraic techniques strategically. In this exercise, the identity \( \cos^4 w + 1 - \sin^4 w = 2 \cos^2 w \) is verified through simplification:

• Translate everything in terms of one or two trigonometric functions, using identities whenever possible.
• Apply algebraic methods like factoring and simplifying.
• Check that both sides of the equation satisfy the simplified forms derived from these steps.

Verification often demands patience and creativity. By converting \( \cos^4 w - \sin^4 w \) into \( \cos^2 w - \sin^2 w \), and showing this equals the simplified right-hand side, we've demonstrated the given identity's validity.
Practice strengthens your skill to recognize when and how to apply these identities correctly, reinforcing a deeper understanding of trigonometric relationships.