Problem 98 - Algebra and Trigonometry

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, \(f(x) = g(x)\) is an identity, as both simplify to \(\cos(2x)\).

Step 1: Understand the functions

We have two functions, \(f(x) = \cos^4 x - \sin^4 x\) and \(g(x) = 2 \cos^2 x - 1\). Our task is to determine if these two functions, when graphed, suggest they are identical, meaning \(f(x) = g(x)\) for all \(x\).

Step 2: Simplify \(f(x)\)

Notice that \(f(x) = \cos^4 x - \sin^4 x\) can be expressed using the identity \(a^2 - b^2 = (a-b)(a+b)\). Thus, \(f(x) = (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x)\). Since \(\cos^2 x + \sin^2 x = 1\), we simplify \(f(x)\) to \(\cos^2 x - \sin^2 x\).

Step 3: Recognize common identities

Recall the identity \(\cos(2x) = \cos^2 x - \sin^2 x\), which matches our expression for \(f(x)\). Therefore, \(f(x) = \cos(2x)\).

Step 4: Simplify \(g(x)\)

The function \(g(x) = 2\cos^2 x - 1\) is another form of a trigonometric identity. It corresponds to \(\cos(2x) = 2\cos^2 x - 1\).

Step 5: Compare \(f(x)\) and \(g(x)\)

Since both \(f(x)\) and \(g(x)\) simplify to the same expression \(\cos(2x)\), they are identical. Hence, \(f(x) = g(x)\) is an identity.

Key Concepts

Function Simplification

Simplifying functions is a fundamental concept in mathematics that helps make complex expressions more manageable. This often involves using mathematical identities or operations to reduce a function to its simplest form. In our original exercise, we have the function

• \( f(x) = \cos^4 x - \sin^4 x \)

At first glance, this expression looks rather complicated, with powers of trigonometric functions. However, an important technique in function simplification is recognizing patterns and identities, such as the difference of squares formula:

• \( a^2 - b^2 = (a-b)(a+b) \)

By applying this identity to \( f(x) \), it becomes:

• \( f(x) = (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x) \)

Since \( \cos^2 x + \sin^2 x \) is a well-known Pythagorean identity equaling 1, \( f(x) \) simplifies to \( \cos^2 x - \sin^2 x \). Identifying and using such identities can hugely simplify the process of working with complex functions. This is a key skill in calculus and higher mathematics.
In this context, simplifying functions leads us to deeper insights and connections between different mathematical expressions.

Trigonometric Functions

Trigonometric functions are a core part of mathematics, especially when dealing with angles and periodic phenomena. The main trigonometric functions—sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \))—represent the basic relationships between the angles and sides of a right-angled triangle.
They also describe periodic waves and oscillations in various fields of study.In our exercise, both

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

are expressed in terms of cosine and sine. The key insight is recognizing these trigonometric functions' identities. For example,

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

These identities show the inherent oscillatory nature of cosine waves, as expressed in terms of double angles. Understanding these properties allows one to manipulate and simplify trigonometric expressions effectively, thereby enhancing problem-solving abilities concerning both theoretical and practical applications.

Identity Verification

Identity verification in mathematics involves proving or confirming that two mathematical expressions are equivalent for all values within their domains. In this exercise, we are asked to verify if

• \( f(x) = \cos^4 x - \sin^4 x \)
• \( g(x) = 2\cos^2 x - 1 \)

are identical, meaning they should satisfy the statement

• \( f(x) = g(x) \)

for all inputs. Simplifying both functions, we have established that each represents \( \cos(2x) \). As both simplify to the same trigonometric identity, we conclude:

• The expressions are indeed identical.

Verification of identities often relies heavily on recognizing fundamental relationships and manipulating them using logical operations and known equations. This approach not only demonstrates the equivalence in a rigorous manner but also builds a deeper understanding of the relationships among mathematical concepts.