Problem 95 - 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. $$ f(x)=\cos ^{2} x-\sin ^{2} x, \quad g(x)=1-2 \sin ^{2} x $$

Step-by-Step Solution

Answer

Yes, \(f(x)=g(x)\) is an identity.

Step 1: Graph the Functions

To begin, we need to graph the functions \(f(x) = \cos^2 x - \sin^2 x\) and \(g(x) = 1 - 2\sin^2 x\). Use a graphing calculator or graphing software to plot both functions over an interval, such as \([0, 2\pi]\). Observe how the graphs overlap and look for any indications that the functions are identical.

Step 2: Analytically Compare the Functions

Check if an identity holds between \(f(x)\) and \(g(x)\) by manipulating the expressions. Recognize the trigonometric identity \(\cos^2 x + \sin^2 x = 1\). We can simplify \(g(x)\) as follows:Substitute \(\cos^2 x = 1 - \sin^2 x\) into \(f(x)\):\[ f(x) = (1 - \sin^2 x) - \sin^2 x = 1 - 2\sin^2 x = g(x). \]

Step 3: Verify through Identity

Since both \(f(x)\) and \(g(x)\) simplify to the same form, they represent the same function, confirming that the equation \(f(x) = g(x)\) is an identity.

Key Concepts

Graphing Trigonometric Functions

When we talk about graphing trigonometric functions, we're essentially visualizing how these functions behave across a certain range of angles. In this exercise, we have two functions, \( f(x) = \cos^2 x - \sin^2 x \) and \( g(x) = 1 - 2 \sin^2 x \), that need to be plotted on the same axis.

To plot these functions, employ a graphing calculator or software, which will show you how these functions vary between 0 and \( 2\pi \) radians. A key insight from plotting these functions is to observe their overlap; this can indicate whether they are identical or not.

For both functions, their periodic nature and repetitions may suggest a correlation. Although different at first glance, both functions will actually unveil themselves to be identical when simplified, which we'll explore further. Visual analysis, combined with analytical verification, ensures a comprehensive understanding of trigonometric identities through this method.

Cosine and Sine Functions

The cosine and sine functions are fundamental to trigonometry and form the basis for understanding more complex expressions. These functions describe the circular motion and relationships within a unit circle.

In this scenario, the roles of cosine and sine are embedded within the expressions \( \cos^2 x\) and \( \sin^2 x\). These squared terms represent the squared projection of a point on the unit circle onto the x-axis and y-axis respectively.

Here, we need to understand the identity \( \cos^2 x + \sin^2 x = 1\). This identity is crucial because it allows us to manipulate and simplify expressions conveniently. By recognizing how cosine and sine work together, we apply this identity to confirm the equality of \( f(x) \) and \( g(x) \). Essentially, understanding how to use these basic functions and identities forms the bedrock of trigonometric simplification.

Trigonometric Simplification

Trigonometric simplification is a key skill when working with trigonometric identities. It involves reducing complex trigonometric expressions into simpler, more manageable forms using known identities.

In our case, to verify that \( f(x) = g(x) \) is indeed an identity, we employ simplification using the identity \( \cos^2 x + \sin^2 x = 1 \). By substituting \( \cos^2 x = 1 - \sin^2 x \) into the expression for \( f(x) \), we discover that \( f(x) \) simplifies to \( 1 - 2\sin^2 x \), which matches \( g(x) \).

By recognizing and applying these identities, you simplify the problem-solving process significantly. Always look for opportunities to apply basic identities to complex expressions, as this will often reveal hidden equivalencies.