Problem 64 - Precalculus with Limits

Question

In Exercises 61-64, verify the identity. \( \sin^4 x + \cos^4 x = 1 - 2 \cos^2 x + 2 \cos^4 x \)

Step-by-Step Solution

Answer

Therefore, the given equation \( \sin^4 x + \cos^4 x \) is indeed equivalent to the equation \( 1 - 2 \cos^2 x + 2 \cos^4 x \) after using the identity \( \sin^2 x + \cos^2 x = 1 \) and simplifying the equations.

Step 1: Breakdown given equation

At first glance, the form \(\sin^4 x + \cos^4 x\) is reminiscent of the square of a binomial (a+b)^2. So, first thing to do is to rewrite it as \((\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x \) . We know that \( \sin^2 x + \cos^2 x = 1 \)

Step 2: Substituting the identity

Next, replace \(\sin^2 x + \cos^2 x\) with 1 in the equation from step 1. Therefore, equation will be \( (1)^2 - 2\sin^2 x \cos^2 x \)

Step 3: Simplify the equation

Simplify the equation to get \( 1 - 2 \sin^2 x \cos^2 x \). But, we realize, it doesn't match the target equation. However we notice that \( \sin^2 x = 1 - \cos^2 x \)

Step 4: Substituting the identity

Next, replace \( \sin^2 x \) with \( 1 - \cos^2 x \) in the equation from step 3. Therefore, equation simplifies to \( 1 - 2 (1 - \cos^2 x) \cos^2 x = 1 - 2\cos^2 x + 2\cos^4 x \) thus proving the identity.

Key Concepts

Verification of Identities

Verifying trigonometric identities can seem daunting at first, but it's like solving a puzzle. The goal is to transform one side of an equation to match the other, using known identities and algebraic manipulation. For the given identity, you begin by transforming the left-hand side,

• Start with \[\sin^4 x + \cos^4 x\]Think of it as parts of an expanded binomial, \[(a + b)^2\]This leads you to rewrite it using another identity.

It's important to recognize that you utilize the identity \[\sin^2 x + \cos^2 x = 1\]. This use of such identities acts as the bridge connecting the two sides of the equation. As the transformation progresses and reaches \[1 - 2\cos^2 x + 2\cos^4 x\], your verification confirms both sides are equal.

Trigonometric Simplification

Simplification plays a pivotal role in verifying trigonometric identities. The process often requires recognizing known identities and rewriting complex expressions with them. In this exercise, the expression \[\sin^4 x + \cos^4 x\]can seem complex due to its powers, but there lurks a pattern resembling a recognizable binomial form.

The crucial step involves expressing \[\sin^4 x + \cos^4 x\]using \[(\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x\]. By using the identity \[\sin^2 x = 1 - \cos^2 x\], you unravel complexities into simpler, known quantities. This process ultimately shows that doing the right substitutions unveils the true equivalence \[1 - 2\cos^2 x + 2\cos^4 x\], proving the initial statement.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry, asserting that \[\sin^2 x + \cos^2 x = 1\]. It is repeatedly used to confirm and simplify expressions, making it an essential tool in verifying identities.

In the current exercise, you substitute \[\sin^2 x + \cos^2 x\] with 1 as a simplifying shortcut, transforming the given expression into a more manageable form. Once \[\sin^2 x\] is swapped for \[1 - \cos^2 x\], the seemingly complex identity becomes simple. Such substitutions capitalize on the consistency and universality of the Pythagorean identity, turning an intricate appearance into an easily understandable equation.

• Key identities facilitate verification.
• Simplification becomes more approachable.
• The identity's powerful role simplifies multi-step transformations.