Problem 46 - 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 Pattern

Look at the left-hand side of the identity, which is \( \sin^4 \theta - \cos^4 \theta \). This can be recognized as a difference of squares, where \( a^2 - b^2 = (a-b)(a+b) \).

Step 2: Apply the Difference of Squares Formula

Rewrite \( \sin^4 \theta - \cos^4 \theta \) using the difference of squares formula:\[\sin^4 \theta - \cos^4 \theta = (\sin^2 \theta)^2 - (\cos^2 \theta)^2 = (\sin^2 \theta - \cos^2 \theta)(\sin^2 \theta + \cos^2 \theta)\]

Step 3: Simplify Using Trigonometric Identity

Recall that \( \sin^2 \theta + \cos^2 \theta = 1 \) (Pythagorean identity). Substitute this into the expression:\[(\sin^2 \theta - \cos^2 \theta)(1)\]

Step 4: Conclude the Verification

Since multiplying by 1 does not change the expression, we have:\[\sin^2 \theta - \cos^2 \theta = \sin^2 \theta - \cos^2 \theta\]Thus, the identity \( \sin^4 \theta - \cos^4 \theta = \sin^2 \theta - \cos^2 \theta \) is verified.

Key Concepts

Difference of Squares

The difference of squares is a mathematical concept that allows us to simplify expressions that involve the subtraction of two squared terms. It's based on the identity where \( a^2 - b^2 = (a-b)(a+b) \). This is a significant tool because it helps decompose complex algebraic expressions into factors, making them easier to manipulate or solve.

In the provided exercise, the expression \( \sin^4 \theta - \cos^4 \theta \) is an example of a difference of squares, where \( a = \sin^2 \theta \) and \( b = \cos^2 \theta \). Applying the identity gives us \( (\sin^2 \theta)^2 - (\cos^2 \theta)^2 = (\sin^2 \theta - \cos^2 \theta)(\sin^2 \theta + \cos^2 \theta) \).

This approach highlights the effectiveness of using the difference of squares to simplify the expression and set the stage for using additional identities for further simplification.

Pythagorean Identity

The Pythagorean identity is one of the fundamental trigonometric identities, stating that \( \sin^2 \theta + \cos^2 \theta = 1 \) for any angle \( \theta \). This identity is derived from the Pythagorean theorem, which relates to the angles of a right triangle.

This identity is particularly useful in simplifying trigonometric expressions, as it provides a constant relationship between the sine and cosine functions. In our exercise, after applying the difference of squares formula, we use the Pythagorean identity to substitute \( \sin^2 \theta + \cos^2 \theta \) with 1, simplifying our expression to \( (\sin^2 \theta - \cos^2 \theta)(1) \).

This step is crucial as it verifies that the expression simplifies correctly by failing to alter the overall value due to multiplication by 1.

Trigonometric Simplification

Trigonometric simplification involves using known identities and algebraic techniques to reduce the complexity of trigonometric expressions. This process often makes it easier to solve equations or prove identities.

In the given exercise, after recognizing and applying the difference of squares, and substituting with the Pythagorean identity, we reach a simplified form. The expression becomes \( \sin^2 \theta - \cos^2 \theta \times 1 \).

Trigonometric simplification confirms our initial goal of proving the identity, establishing that the complex expression on the left-hand side equals the more straightforward expression on the right-hand side. This skill is vital in advanced trigonometry, as it aids in proving more involved identities and solving trigonometric equations efficiently.