Problem 100 - Precalculus Mathematics for Calculus

Question

Graph \(f\) and \(g\) in the same viewing rectangle. Do the graphs suggest that the equation \(f(x)=g(x)\) is an identity? Prove your answer. $$f(x)=\cos ^{4} x-\sin ^{4} x, \quad g(x)=2 \cos ^{2} x-1$$

Step-by-Step Solution

Answer

Yes, the graphs suggest they are identical. The equation \( f(x) = g(x) \) is an identity.

Step 1: Understand the Functions

First, let's identify and understand the two functions. We have two functions given: \( f(x) = \cos^4 x - \sin^4 x \) and \( g(x) = 2\cos^2 x - 1 \). Our task is to graph these functions and analyze if they suggest that \( f(x) = g(x) \) is an identity, meaning they are equal for all values of \(x\).

Step 2: Simplify Function f(x)

We can simplify the expression for \( f(x) = \cos^4 x - \sin^4 x \) using an algebraic identity \(a^2 - b^2 = (a-b)(a+b)\). Here, let \( a = \cos^2 x \) and \( b = \sin^2 x \), then \( f(x) = (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x) \). Since \( \cos^2 x + \sin^2 x = 1 \) for all \(x\), we have \( f(x) = \cos^2 x - \sin^2 x \).

Step 3: Compare f(x) and g(x)

After simplification, observe that \( f(x) = \cos^2 x - \sin^2 x \) and \( g(x) = 2\cos^2 x - 1 \). The identity \( \cos^2 x = 1 - \sin^2 x \) helps us recognize that \( \cos^2 x - \sin^2 x = (1-\sin^2 x) - \sin^2 x = 1-2\sin^2 x \). Thus, \( f(x) = 2\cos^2 x - 1 = g(x) \).

Step 4: Graph the Functions

Graph \( f(x) \) and \( g(x) \) using a graphing tool or by hand. Since we found \( f(x) \) equals \( g(x) \) algebraically, their graphs should overlap completely, reinforcing the notion that the equation \( f(x) = g(x) \) is an identity.

Step 5: Conclusion on Identity

Given that both functions simplify to the same expression and their graphs overlap, we conclude that \( f(x) = g(x) \) is indeed an identity. Hence, both expressions represent the same function over the domain of \( x \).

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions can be a fascinating exploration into the world of oscillations and waves. In this exercise, you're asked to graph two functions: \( f(x) = \cos^4 x - \sin^4 x \) and \( g(x) = 2\cos^2 x - 1 \). Before graphing, understanding their behavior helps. Both functions are derived from basic trigonometric identities and exhibit periodic behavior.
When graphing functions like \( f(x) \) and \( g(x) \), it's crucial to pay attention to the periodicity and symmetry of the trigonometric components.

• The function \( \cos x \) has a period of \( 2\pi \), which means it repeats every \( 2\pi \).
• The same applies to \( \sin x \).

For these functions, the variations in power and coefficients affect the amplitude and potentially the period. However, for these specific cases, the primary period remains tied to \( 2\pi \).
Once graphed, these functions illuminate a key insight: identifying if they coincide across all their intervals validates if their expressions represent the same identity.

Algebraic Simplification

Algebraic simplification is a powerful tool that helps us uncover underlying relationships between mathematical expressions. In this problem, simplifying \( f(x) = \cos^4 x - \sin^4 x \) plays a crucial role in identifying its equivalence with \( g(x) \).
We utilize the identity \( a^2 - b^2 = (a-b)(a+b) \) to simplify \( f(x) \). By setting \( a = \cos^2 x \) and \( b = \sin^2 x \), we transform the expression:

• \( f(x) = (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x) \)
• Since \( \cos^2 x + \sin^2 x = 1 \), this simplifies to \( f(x) = \cos^2 x - \sin^2 x \).

Recognizing that \( \cos^2 x = 1 - \sin^2 x \) enables further simplification:

• \( f(x) = (1 - \sin^2 x) - \sin^2 x = 1 - 2\sin^2 x \)

This simplification reveals that \( f(x) \) is identical to the expression for \( g(x) \), emphasizing the power of algebra in linking seemingly different expressions.

Function Comparison

Comparing functions involves understanding how each function behaves across its domain and determining if they exhibit identical characteristics. In this case, you have \( f(x) = \cos^4 x - \sin^4 x \) and \( g(x) = 2\cos^2 x - 1 \). Simplifying \( f(x) \) to match \( g(x) \) allows for a clear comparison.
Once simplified, we see both functions express \( f(x) = g(x) = 2\cos^2 x - 1 \). This implies their graphs should perfectly overlap, establishing them as functionally equivalent for any input \( x \).
Function comparison is crucial when proving identities; this step validates that different initial expressions can manifest as the same function on a graph, confirming with visual and algebraic consistency. Ultimately, thorough comparison reassures us that despite initial complexities, the functions fundamentally coincide.