Problem 42 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\cos ^{4} w+1-\sin ^{4} w=2 \cos ^{2} w$$

Step-by-Step Solution

Answer

The identity is verified as both sides equal \( 2\cos^{2} w \).

Step 1: Understanding the Question

We have an expression on the left side \( \cos^{4} w + 1 - \sin^{4} w \) and need to verify if it is equal to \( 2 \cos^{2} w \). Our goal is to make transforms and simplifications that show both sides are equal.

Step 1: Utilize Pythagorean Identity

Recall the Pythagorean Identity: \( \cos^{2} w + \sin^{2} w = 1 \). Use this to express \( \sin^{2} w \) in terms of \( \cos^{2} w \): \( \sin^{2} w = 1 - \cos^{2} w \). Substitute into the equation for \( \sin^{4} w \): \( \sin^{4} w = (1 - \cos^{2} w)^2 \).

Step 2: Expand \( \sin^{4} w \)

Expand \( (1 - \cos^{2} w)^2 = 1 - 2\cos^{2} w + \cos^{4} w \). Now substitute back into the original equation so it becomes: \( \cos^{4} w + 1 - (1 - 2\cos^{2} w + \cos^{4} w) \).

Step 3: Simplify the Equation

Simplify the expression: \( \cos^{4} w + 1 - 1 + 2\cos^{2} w - \cos^{4} w \). This simplifies to \( 2\cos^{2} w \).

Step 5: Conclusion: Verify the Identity

Both sides of the equation simplify to \( 2\cos^{2} w \). Therefore, the identity \( \cos^{4} w + 1 - \sin^{4} w = 2 \cos^{2} w \) is verified.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a cornerstone in trigonometry. It states that for any angle \( w \), the sum of the square of the cosine and the square of the sine is always equal to one, given by: \[ \cos^2 w + \sin^2 w = 1 \]This fundamental relationship is derived from the Pythagorean theorem that you use when working with right-angled triangles. Think of it as a handy tool that helps to find one trigonometric function when you know the other.
In our exercise, the identity is used to rewrite \( \sin^2 w \) in terms of \( \cos^2 w \). This substitution forms the basis for further steps of simplification and verification. Using this identity, you can transform one part of a trigonometric equation into something often easier to work with or match the other side.

• Convert \( \sin^2 w \) as: \( \sin^2 w = 1 - \cos^2 w \)
• This lets us replace \( \sin^4 w \) with \( (1 - \cos^2 w)^2 \), a crucial alternate expression for simplifying the given identity.

Trigonometric Simplification

Trigonometric simplification involves reducing complex expressions into more manageable forms. It often requires applying identities and algebraic techniques to achieve the desired simplification.
In the provided exercise, simplifying involves expanding \( (1 - \cos^2 w)^2 \). To break down the expression, we use methods like the distributive property (often referred to as "FOIL" when working with binomials) to achieve: \[ (1 - \cos^2 w)^2 = 1 - 2\cos^2 w + \cos^4 w \]This aligns different parts of the trigonometric expression, making it easier to simplify further.
Next, by applying changes back into the original equation, the simplification results in: \[ \cos^{4} w + 1 - \sin^{4} w = 2 \cos^{2} w \]Simplifying these expressions step-by-step is like peeling back layers of complexity until you arrive at a form where you can easily see the equivalence or identity.

• Understand that multiplying terms and expanding them allows for eliminating and combining like terms.
• The key is transforming and rearranging terms so they are seen clearly in their simplest form possible.

Equivalent Expressions

Equivalent expressions are different forms of the same mathematical entity. In trigonometry, showing two expressions to be equivalent typically involves verifying or proving identities. An identity is an equation that holds true for all values of the variable involved.
In this exercise, we started with the expression \( \cos^{4} w + 1 - \sin^{4} w \) and transformed it through simplification into \( 2 \cos^{2} w \), confirming both sides are equal, thus proving the trigonometric identity.
Understanding equivalency means recognizing that different-looking expressions can represent the same numerical value. It encapsulates the flexibility of mathematical expressions, allowing them to be manipulated to reach desired forms or solutions.

• Recognize that expressions can be official identities or temporary formats for solving particular problems.
• In the context of verifying identities, each transformation is a deliberate step to show equivalency.

Being adept at recognizing and proving equivalence fosters deeper understanding and application of mathematical principles in solving problems.