Problem 96 - 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. $$ f(x)=\tan x(1+\sin x), \quad g(x)=\frac{\sin x \cos x}{1+\sin x} $$

Step-by-Step Solution

Answer

The equation \( f(x) = g(x) \) is not an identity.

Step 1: Understanding the Functions

The functions given are \( f(x) = \tan x (1+\sin x) \) and \( g(x) = \frac{\sin x \cos x}{1+\sin x} \). We need to graph these functions and see if they are identical for all \(x\).

Step 2: Simplifying f(x)

The function \( f(x) = \tan x (1+\sin x) \) can be rewritten using \( \tan x = \frac{\sin x}{\cos x} \). Hence, \( f(x) = \frac{\sin x}{\cos x} (1+\sin x) = \frac{\sin x (1+\sin x)}{\cos x} \).

Step 3: Simplifying g(x)

The function \( g(x) = \frac{\sin x \cos x}{1+\sin x} \) is already simplified, as it expresses \( \,\) a fraction directly using trigonometric values.

Step 4: Analyzing for Equality

To determine if \( f(x) = g(x)\) is an identity, we need to see if it holds for all \(x\) where both are defined. Equate \( f(x) = \frac{\sin x (1+\sin x)}{\cos x} \) and \( g(x) = \frac{\sin x \cos x}{1+\sin x} \).

Step 5: Cross-multiplying and Simplifying

Upon cross-multiplying, we have \(\sin x (1+\sin x)(1+\sin x) = \cos^2 x \sin x\). Simplify the equation as \((1+\sin x)^2 = \cos^2 x\).

Step 6: Using the Pythagorean Identity

Using the trigonometric identity \( \sin^2 x + \cos^2 x = 1 \), rearrange to find \( 1+2\sin x+\sin^2 x = 1-\sin^2 x \). This gives \( 2\sin^2 x + 2\sin x = 0\).

Step 7: Solving the Identity Equation

Factor out \( 2\sin x \) resulting in \( 2\sin x(\sin x + 1) = 0 \). This only holds for specific \(x\) where \(\sin x = 0\) or \(\sin x = -1\), not for all \(x\).

Step 8: Conclusion from Graph and Algebra

Since the algebraic simplification shows that the condition of equality is not satisfied for all \(x\), \( f(x) eq g(x) \) for all \(x\), except at specific points. Thus, it is not an identity.

Key Concepts

Graphing Functions

When graphing functions such as \(f(x) = \tan x (1+\sin x)\) and \(g(x) = \frac{\sin x \cos x}{1+\sin x}\), a critical part of the process is understanding what each function represents. Graphs provide a visual perspective that can help determine relationships and characteristics between different functions. To graph these functions correctly, it is essential to choose an appropriate viewing window that captures the behavior of both functions, including periodicity and any asymptotes or discontinuities.

• Identifying Characteristics: Look for periodic behavior, points of discontinuity, or undefined areas for each function due to trigonometric properties.


• Comparison: Place both graphs on the same axis to directly observe if they coincide or differ at any points.

Such visualization helps in hypothesizing whether \(f(x) = g(x)\) is a trigonometric identity or not, as an identity implies the graphs should perfectly overlap for all values within the domain.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is fundamental when trying to compare them by algebraic means. In our scenario with \(f(x) = \frac{\sin x (1+\sin x)}{\cos x}\) and \(g(x) = \frac{\sin x \cos x}{1+\sin x}\), we aim to break them down using known identities to see if they can be expressed similarly.

• Breaking Down Functions: Use known trigonometric identities like \(\tan x = \frac{\sin x}{\cos x}\) to rewrite them in simpler forms. Recognizing these patterns can vastly reduce complexity.


• Rewriting Aim: Transform expressions to be easily comparable or equatable.

By simplifying, we can focus on fundamental components of the expressions to make direct comparisons, essential in showing if the functions are indeed identical.

Pythagorean Identity

The Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) serves as a cornerstone in simplifying complex trigonometric equations. This identity is instrumental in restructuring expressions during simplification or solving tasks.

• Application in Problem: Used to rearrange terms and expose possible relations between functions. For instance, twisting \(1 - \sin^2 x\) as \(\cos^2 x\) helped to expose a square form for simplification.


• Realization of Context: It aids in verifying identities and simplifying expressions to expose solutions not visible through regular reduction techniques.

Understanding and applying different forms of this identity lets you access a larger toolset to solve or graph complicated trigonometric relations.

Cross-Multiplying Equations

Cross-multiplying is a powerful algebraic technique often used to solve equations involving fractions. When comparing two functions like \(f(x)\) and \(g(x)\), cross-multiplying can help eliminate denominators, making the expressions easier to manipulate and compare.

• Eliminating Denominators: By multiplying both sides of an equation by the denominators, you effectively remove fractions, simplifying comparison between expressions.


• Simplification: This deals with expressions in a more straightforward form. Once simplified, the final equation can reveal whether the original functions are equivalent or highlight specific values where they might intersect.

Through cross-multiplying in this problem, we reached a factored form revealing that \(f(x) = g(x)\) only holds at specific points, thus proving it isn't a complete identity across all \(x\).