Problem 97 - 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 because they simplify to the same expression.

Step 1: Understand the Functions

Identify the given functions, \(f(x) = \cos^2 x - \sin^2 x\) and \(g(x) = 1 - 2 \sin^2 x\). These functions might be trigonometric identities.

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

Use the trigonometric identity \(\cos^2 x = 1 - \sin^2 x\) so that \(f(x)\) can be rewritten as follows:\[ f(x) = (1 - \sin^2 x) - \sin^2 x = 1 - 2\sin^2 x \]

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

From Step 2, we found that \[ f(x) = 1 - 2\sin^2 x \] This is identical to the expression we have for \(g(x)\), i.e., \(g(x) = 1 - 2\sin^2 x\).

Step 4: Conclude the Identity

Since both expressions for \(f(x)\) and \(g(x)\) result in the same form, the equation \(f(x) = g(x)\) holds for any value of \(x\). This means \(f(x) = g(x)\) is an identity.

Key Concepts

Cosine Function

The cosine function, denoted as \(\cos(x)\), is one of the fundamental trigonometric functions. It relates the angle of a right triangle to the ratio of the adjacent side over the hypotenuse. Given its periodic nature, cosine is crucial for describing oscillatory motion and wave-like behavior. For example, it helps explain phenomena like sound waves or the motion of a pendulum.
In our original exercise, cosine is squared, with the function defined as \(\cos^2 x\). Squaring the cosine function can be perceived as finding the area of a square with sides equal to \(\cos(x)\). This transformation maintains positive values, amplifying the cosine's unique properties. In the identity \(f(x) = \cos^2 x - \sin^2 x\), cosine's role is vital in forming the structure of the equation.
It's crucial to remember the important identity:

• \(\cos^2 x + \sin^2 x = 1\)

This identity allows for simplification by expressing \(\cos^2 x\) in terms of \(\sin^2 x\), which was crucial in proving \(f(x) = g(x)\). Understanding this identity sets the foundation for manipulating and simplifying trigonometric expressions.

Sine Function

The sine function, represented as \(\sin(x)\), is another key trigonometric function. It relates an angle with the ratio of the opposite side over the hypotenuse in a right triangle. Just like the cosine function, the sine function is periodic and oscillates between -1 and 1. It's instrumental in modelling cyclic phenomena ranging from alternating current to heartbeats.
In the given exercise, \(\sin^2 x\) appears in both functions \(f(x)\) and \(g(x)\). When we talk about \(\sin^2 x\), we refer to squaring the sine value for an angle, which results in non-negative values between 0 and 1. This characteristic is critical when performing the simplification tasks in trigonometric identities.
The sine identity combined with cosine creates the fundamental relationship:

• \(\cos^2 x + \sin^2 x = 1\)

This helps in transforming the expression \(\cos^2 x\) to \(1 - \sin^2 x\) and vice versa. Such transformations allow for simplification and provide insight into trigonometric relationships, as seen in our function simplification from the exercise.

Trigonometric Simplification

Trigonometric simplification is a tool to make complex trigonometric expressions easier to handle. This is achieved using known identities and transformations between different trigonometric functions. Simplification often helps us understand relationships between functions better and can reveal hidden identities.
In the exercise, simplification of \(f(x) = \cos^2 x - \sin^2 x\) was key. Using the identity \(\cos^2 x = 1 - \sin^2 x\), we turned \(f(x)\) into the same form as \(g(x) = 1 - 2 \sin^2 x\). Here's a simple breakdown of the steps:

• Recognize the identity \(\cos^2 x = 1 - \sin^2 x\).
• Substitute this identity into \(f(x)\) to get \((1 - \sin^2 x) - \sin^2 x\).
• Combine terms to solve \(f(x) = 1 - 2\sin^2 x\).

Through this process, we see how aligning terms with their trigonometric identities brings clarity and solves complex equations. Without simplification techniques, students might miss the equivalences that identities represent, such as confirming \(f(x) = g(x)\) in our exercise. Effective simplification aids in better understanding and problem-solving using trigonometric identities.