Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Simplify the expression.$$ \frac{\left(e^{x}+e^{-x}\right)\left(e^{x}+e^{-x}\right)-\left(e^{x}-e^{-x}\right)\left(e^{x}-e^{-x}\right)}{\left(e^{x}+e^{-x}\right)^{2}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{4}{e^{2x} + 2 + e^{-2x}} \).

Step 1: Expand the Numerator

The expression is \( (e^x + e^{-x})^2 - (e^x - e^{-x})^2 \). Use the formula \((a+b)^2 = a^2 + 2ab + b^2\) to expand \((e^x + e^{-x})^2\) and \((a-b)^2 = a^2 - 2ab + b^2\) to expand \((e^x - e^{-x})^2\). This gives: \[ (e^x + e^{-x})^2 = e^{2x} + 2 + e^{-2x} \] \[ (e^x - e^{-x})^2 = e^{2x} - 2 + e^{-2x} \].

Step 2: Simplify the Expanded Terms

Subtract the second expansion from the first:\[ (e^x + e^{-x})^2 - (e^x - e^{-x})^2 = (e^{2x} + 2 + e^{-2x}) - (e^{2x} - 2 + e^{-2x}) \] By cancellation, \[ = e^{2x} + 2 + e^{-2x} - e^{2x} + 2 - e^{-2x} = 4 \].

Step 3: Simplify the Denominator

The denominator of the expression is \((e^x + e^{-x})^2\) which we have previously expanded: \[ = e^{2x} + 2 + e^{-2x} \].

Step 4: Combine the Result

Now, simplify the overall expression by dividing the simplified numerator by the denominator:\[ \frac{4}{e^{2x} + 2 + e^{-2x}} \]. This is already in its simplest form.

Key Concepts

Algebraic Identities

Algebraic identities are mathematical statements that equate two algebraic expressions for any possible values of their variables. In the exercise above, the identities

• \((a+b)^2 = a^2 + 2ab + b^2\)
• \((a-b)^2 = a^2 - 2ab + b^2\)

are used to expand the expressions \((e^x + e^{-x})^2\) and \((e^x - e^{-x})^2\).
These identities simplify calculations because they provide a standard form to break down and solve complex expressions. For example, instead of independently multiplying \((e^x + e^{-x})\) by itself, the identity gives a shortcut by immediately providing the result.
Recognizing and applying these identities can help simplify expressions quickly and accurately, making them an essential tool in algebra.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In this exercise, the base \(e\) is raised to the power of both positive and negative \(x\), which are common expressions in exponential functions.

• The function \(e^x\) describes exponential growth, where the value increases as \(x\) increases.
• Conversely, \(e^{-x}\) represents exponential decay, decreasing as \(x\) grows.

Understanding the behavior of these functions is crucial in algebra, as they often require simplification and manipulation to solve complex equations.
When simplifying the original expression, knowing how to handle \(e^x\) and \(e^{-x}\) is key, especially when you see them squared or combined in various forms. Recognizing these patterns can make working with exponential expressions more intuitive and manageable.

Rational Expressions

Rational expressions are quotients of two polynomials or similar algebraic expressions. In the exercise, the original equation takes the form of a rational expression: \[\frac{(e^{x}+e^{-x})^2-(e^{x}-e^{-x})^2}{(e^{x}+e^{-x})^2}\] Such expressions require careful simplification, focusing on what can be canceled or reduced.
The solution begins by expanding the numerator and recognizing common terms that can be simplified. The rational expression simplifies to \(\frac{4}{e^{2x} + 2 + e^{-2x}}\).
The ability to simplify rational expressions is vital for solving algebraic problems, as it often leads to discovering the simplest form of an equation, making further calculations clear and straightforward.