Problem 84 - A Graphical Approach to Precalculus with Limits

Question

Verify that equation is an identity. \(\frac{\sin ^{4} \alpha-\cos ^{4} \alpha}{\sin ^{2} \alpha-\cos ^{2} \alpha}=1\)

Step-by-Step Solution

Answer

The expression simplifies to 1, verifying the identity.

Step 1: Recognize the Form of the Given Expression

The given expression is \( \frac{\sin^4 \alpha - \cos^4 \alpha}{\sin^2 \alpha - \cos^2 \alpha} \). Notice that both the numerator and the denominator remind us of common algebraic identities, particularly the difference of squares.

Step 2: Apply the Difference of Squares Identity

Recall that \( a^2 - b^2 = (a-b)(a+b) \). Therefore, \( \sin^4 \alpha - \cos^4 \alpha \) can be rewritten as a difference of squares: \( (\sin^2 \alpha)^2 - (\cos^2 \alpha)^2 = (\sin^2 \alpha - \cos^2 \alpha)(\sin^2 \alpha + \cos^2 \alpha) \).

Step 3: Simplify the Expression

Substitute the factorized form of the numerator into the expression: \[ \frac{(\sin^2 \alpha - \cos^2 \alpha)(\sin^2 \alpha + \cos^2 \alpha)}{\sin^2 \alpha - \cos^2 \alpha} \]Cancel out the common term \( (\sin^2 \alpha - \cos^2 \alpha) \) in the numerator and denominator, assuming \( \sin^2 \alpha eq \cos^2 \alpha \): \[ \sin^2 \alpha + \cos^2 \alpha \].

Step 4: Recognize a Pythagorean Identity

Recall the fundamental Pythagorean identity: \( \sin^2 \alpha + \cos^2 \alpha = 1 \). Using this identity, simplify the expression to 1.

Key Concepts

Difference of Squares

In algebra, the difference of squares is a fundamental identity that is used to simplify expressions. It states that for any two numbers, \( a \) and \( b \), the expression \( a^2 - b^2 \) can be rewritten as \( (a-b)(a+b) \). This identity is useful when needing to factorize expressions that fit this form.
For the original exercise, we use the difference of squares to transform \( \sin^4 \alpha - \cos^4 \alpha \) into \( (\sin^2 \alpha)^2 - (\cos^2 \alpha)^2 \). This expression fits the difference of squares form, allowing us to rewrite it as:

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

This transformation simplifies the problem significantly because it allows us to cancel common terms later on.

Pythagorean Identity

The Pythagorean identity is a central concept in trigonometry. At its heart, it simply states that for any angle \( \alpha \), the sum of the squares of sine and cosine is always one:

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

In the context of the problem, after applying the difference of squares, we simply ended up with an expression \( \sin^2 \alpha + \cos^2 \alpha \) which is precisely the Pythagorean identity.
By recognizing this, we can immediately substitute it with 1, simplifying the whole expression to equal 1. Understanding and applying this identity is crucial in verifying trigonometric identities efficiently.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to achieve a desired form. It starts with identifying patterns and using known identities or properties.
In the provided solution, algebraic manipulation began with recognizing that the original expression could be broken down by the difference of squares. After factorizing the expression, it involved canceling out common terms:

• The term \( (\sin^2 \alpha - \cos^2 \alpha) \) appears in both the numerator and the denominator, enabling cancellation.

This gave us a simpler expression of \( \sin^2 \alpha + \cos^2 \alpha \), which was then simplified using the Pythagorean identity.
This process showcases the power of algebraic manipulation in simplifying complex trigonometric identities.