Problem 48 - Algebra and Trigonometry

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 using the difference of squares and the Pythagorean identity.

Step 1: Recognize the identity form

The left side of the equation is structured as a difference of squares, \( a^2 - b^2 \), where \( a = \sin^2 \theta \) and \( b = \cos^2 \theta \). Recall the formula: \( a^2 - b^2 = (a - b)(a + b) \).

Step 2: Apply the difference of squares formula

Rewrite the left side of the identity using the difference of squares formula. Substitute \( a = \sin^2 \theta \) and \( b = \cos^2 \theta \) to get:\[(\sin^2 \theta - \cos^2 \theta)(\sin^2 \theta + \cos^2 \theta)\]

Step 3: Simplify the equation

We know from the Pythagorean identity that \( \sin^2 \theta + \cos^2 \theta = 1 \).Replace this in the expression obtained from Step 2, resulting in:\[(\sin^2 \theta - \cos^2 \theta)(1) = \sin^2 \theta - \cos^2 \theta\]

Step 4: Verify the original identity

Both sides of the equation simplify to \( \sin^2 \theta - \cos^2 \theta \), which shows that the identity holds true. Therefore, the original expression \( \sin^4 \theta - \cos^4 \theta = \sin^2 \theta - \cos^2 \theta \) is verified.

Key Concepts

Difference of Squares

The concept of the difference of squares is a useful algebraic identity that helps simplify expressions involving two squared terms. When you see the structure \( a^2 - b^2 \), recognize it as a perfect candidate for the difference of squares formula: \( (a - b)(a + b) \).
This formula breaks the expression into the product of two binomials:

• \( (a - b) \) is the first factor, involving the difference between the two terms.
• \( (a + b) \) is the second factor, showcasing the sum of the terms.

In our exercise, this formula transforms \( \sin^4 \theta - \cos^4 \theta \) into \( (\sin^2 \theta - \cos^2 \theta)(\sin^2 \theta + \cos^2 \theta) \). This simplification is crucial, as it allows us to use further trigonometric identities to continue the simplification.

Pythagorean Identity

The Pythagorean Identity is one of the fundamental properties of trigonometry. It states that for any angle \( \theta \), the sum of the square of sine and the square of cosine equals one. Formally, it's expressed as:

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

This identity is highly reliable and serves as a cornerstone for simplifying trigonometric expressions. In our exercise, once the expression \( (\sin^2 \theta + \cos^2 \theta) \) appears, we can replace it with \( 1 \) due to the Pythagorean Identity.
This replacement simplifies the expression down to just \( (\sin^2 \theta - \cos^2 \theta) \), making it easier to verify the given identity.

Trigonometric Simplification

Trigonometric simplification often involves using various identities and algebraic techniques to make an expression more manageable. In our exercise, we utilized the difference of squares to break down the left side of the equation and then applied the Pythagorean Identity.
These steps reduced a seemingly complex equation into something that matches the right side: \( \sin^2 \theta - \cos^2 \theta \). To achieve simplification:

• Identify patterns that resemble known trigonometric identities or algebraic formulas.
• Substitute known identities, like the Pythagorean Identity, whenever possible.
• Aim to transform the expression into a more recognizable or simplified form.

By strategically applying these steps, we not only verify the identity but also practice fundamental trigonometric manipulations useful in more advanced problems.