Problem 48 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The identity is verified by expressing as a difference of squares and using the Pythagorean identity.

Step 1: Express the Left Side

Let's start by expressing the left side of the given identity, \(\sin^4 \theta - \cos^4 \theta\), as a difference of squares. We know that \(a^2 - b^2 = (a - b)(a + b)\). In this case, \(a = \sin^2 \theta\) and \(b = \cos^2 \theta\), so we have:\[\sin^4 \theta - \cos^4 \theta = (\sin^2 \theta - \cos^2 \theta)(\sin^2 \theta + \cos^2 \theta)\]

Step 2: Simplify Using the Pythagorean Identity

Now, apply the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute this into the expression we found in Step 1:\[(\sin^2 \theta - \cos^2 \theta)(\sin^2 \theta + \cos^2 \theta) = (\sin^2 \theta - \cos^2 \theta)(1)\]

Step 3: Final Equation

Now, simplifying the expression from Step 2, we are left with:\[\sin^4 \theta - \cos^4 \theta = \sin^2 \theta - \cos^2 \theta\]This expression matches the right side of the given identity, verifying that the original identity is correct.

Key Concepts

Difference of Squares

The concept of the difference of squares is a powerful algebraic tool used to simplify expressions such as \(a^2 - b^2\). The formula states \((a^2 - b^2) = (a - b)(a + b)\). This works because multiplying \((a - b)\) by \((a + b)\) results in \(a^2 - b^2\) after applying the distributive property and subtracting the \(ab\) terms.In our exercise, the left side of the given trigonometric identity is \(\sin^4 \theta - \cos^4 \theta\). By identifying this expression as a difference of squares, we let \(a = \sin^2 \theta\) and \(b = \cos^2 \theta\). Therefore, the expression becomes

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

This factorization is a crucial step in simplifying the expression and is the foundation for further manipulation.

Pythagorean Identity

The Pythagorean identity is one of the fundamental trigonometric identities. It states that for any angle \(\theta\), the sum of the squares of sine and cosine is always equal to 1. Mathematically, this is expressed as:

• \(\sin^2 \theta + \cos^2 \theta = 1\)

This identity stems from the Pythagorean theorem, hence its name, and it is pivotal in simplifying trigonometric expressions.In the context of the exercise, after applying the difference of squares, we obtained \((\sin^2 \theta + \cos^2 \theta)\) as part of the expression. Using the Pythagorean identity, we can simplify this term to 1. This step significantly reduces the complexity of the given problem and helps verify the trigonometric identity. Thus, substituting \(\sin^2 \theta + \cos^2 \theta = 1\) allows us to simplify our expression efficiently.

Trigonometric Simplification

Trigonometric simplification involves the process of reducing complex trigonometric expressions to simpler forms, often using identities like the Pythagorean identity, angle sum and difference identities, or factoring techniques. This skill is essential for verifying or transforming expressions in trigonometry.In this exercise, after applying the difference of squares and the Pythagorean identity, we ended up with the term \((\sin^2 \theta - \cos^2 \theta)(1)\). The multiplication by 1 does not change the term, thereby simplifying the expression directly to \(\sin^2 \theta - \cos^2 \theta\).Simplifying expressions can make them easier to handle and understand, especially when verifying identities. The process of breaking down complex expressions into known identities is a key technique often required to match both sides of an equation or to find a solution. In our exercise, the step-by-step simplification confirmed that \(\sin^4 \theta - \cos^4 \theta\) equals \(\sin^2 \theta - \cos^2 \theta\), thus verifying the identity effectively.