Problem 100 - 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, the equation \(f(x) = g(x)\) is an identity.

Step 1: Analyze Trigonometric Identities

Let's start by using trigonometric identities to simplify both functions. We recognize the difference of squares: \[\cos^4x - \sin^4x = (\cos^2x + \sin^2x)(\cos^2x - \sin^2x)\]Using the identity \(\cos^2x + \sin^2x = 1\), the expression simplifies to:\[\cos^4x - \sin^4x = \cos^2x - \sin^2x \]

Step 2: Simplify Function Expressions

Now, we have the simplified expression for \(f(x)\):\[f(x) = \cos^2x - \sin^2x\]Next, notice that \(g(x)\) is given by the double angle identity for cosine:\[ g(x) = 2\cos^2x - 1 \]Also, recall the identity:\[2\cos^2x - 1 = \cos(2x)\]which will be essential for comparison.

Step 3: Compare Simplified Expressions

We now have:\[ f(x) = \cos^2x - \sin^2x \]and:\[ g(x) = 2\cos^2x - 1 = \cos(2x) \]Using the identity \(\cos(2x) = \cos^2x - \sin^2x\), we find:\[f(x) = \cos(2x)\]This shows \(f(x) = g(x)\) for all \(x\).

Step 4: Verify Graphically

Graph both functions in the same viewing window. You'll see that both functions are identical, confirming that the relationship \(f(x) = g(x)\) holds universally as an identity over all real \(x\).

Step 5: Conclusion

Since both algebraic and graphical methods confirm that \(f(x)\) and \(g(x)\) are equal for all \(x\), the equation \(f(x) = g(x)\) is indeed an identity.

Key Concepts

Graphing Functions

Graphing functions can be a powerful way to visually compare and verify mathematical equations. In this exercise, we were tasked with graphing two functions, \(f(x)\) and \(g(x)\), to see if they appear identical. When graphing functions:

• Start by understanding the behavior of each function separately. Identify their key features, such as amplitude, frequency, and zeros.
• Use a consistent viewing window to display both functions, ensuring they can be compared directly.
• Look for matching shapes, such as overlapping waves or similar periodic behaviors.

For \(f(x) = \cos^4x - \sin^4x\) and \(g(x) = 2\cos^2x - 1\), both functions, when graphed together, overlap perfectly. This graphical equivalence supports the identity suggested by the algebraic simplification.

Trigonometric Functions

Trigonometric functions, like sine and cosine, are foundational in mathematics, especially in periodic phenomena such as waves. They help define relationships in circles and angles. Understanding trigonometric functions involves:

• Knowing their basic shapes: Sine and cosine functions oscillate between -1 and 1.
• Understanding key properties: Periodicity, amplitude, and phase shifts.
• Utilizing identities: Relationships like \(\cos^2x + \sin^2x = 1\) simplify expressions.

In our exercise, these functions form the basis of our identities. By understanding the fundamental properties of cosine, we were able to simplify \(\cos^4x - \sin^4x\) into \(\cos^2x - \sin^2x\), revealing it as a simpler trigonometric function matching \(g(x)\).

Difference of Squares

The difference of squares is a powerful algebraic tool used to simplify certain expressions. Given two terms \(a\) and \(b\), the difference of their squares is expressed as \(a^2 - b^2 = (a+b)(a-b)\). Here’s how to use this concept effectively:

• Identify expressions that fit the \(a^2 - b^2\) pattern.
• Factor them into two simpler binomial expressions.
• Use identities to further simplify or transform these expressions.

In the specific case of \(\cos^4x - \sin^4x\), recognizing this as a difference of squares allows us to factor and use trig identities, leading to simpler equivalent expressions, aiding in confirming functional identities.

Cosine Double Angle Identity

The cosine double angle identity is a key trigonometric formula that simplifies expressions involving \(\cos(2x)\). The identity is:\[ \cos(2x) = 2\cos^2x - 1 \]. This identity helps to express the cosine of a double angle using the square of cosine. Using this identity involves:

• Recognizing expressions that fit or can be transformed into \(\cos(2x)\).
• Simplifying complex trigonometric expressions by substitution.
• Verifying identities and solving equations involving double angles.

In our exercise, \(g(x) = 2\cos^2x - 1\) is directly given by this identity, allowing easy comparison to the simplified \(f(x) = \cos(2x)\), confirming their equality.