## Exploring the Expression (x² - y²) / (x² + y²)

The expression (x² - y²) / (x² + y²) is a simple yet intriguing algebraic expression. It can be analyzed and manipulated in several ways, leading to interesting insights.

### Factoring and Simplifying

One of the first things we can do is factor the numerator and denominator:

**Numerator:** (x² - y²) factors into (x + y)(x - y) (difference of squares)

**Denominator:** (x² + y²) does not factor further in the real number system.

This gives us:

**(x² - y²) / (x² + y²) = (x + y)(x - y) / (x² + y²)**

While this factored form isn't necessarily simpler, it highlights the underlying structure of the expression and can be useful in various contexts.

### Understanding the Expression's Behavior

To understand the expression's behavior, we can analyze it for different values of x and y:

**When x = y:**The numerator becomes zero, and the entire expression evaluates to**0**.**When x = 0:**The expression simplifies to**-1**.**When y = 0:**The expression simplifies to**1**.

This shows that the expression can take on a wide range of values depending on the values of x and y.

### Relation to Trigonometry

Interestingly, the expression (x² - y²) / (x² + y²) can be connected to trigonometric functions. Consider a right triangle with sides x, y, and hypotenuse z. Using the Pythagorean theorem (z² = x² + y²), we can rewrite the expression as:

**(x² - y²) / (x² + y²) = (x² - y²) / z²**

Now, dividing both numerator and denominator by z², we get:

**(x² / z²) - (y² / z²) = (x / z)² - (y / z)²**

Remembering that x/z = cos θ and y/z = sin θ (where θ is the angle opposite side y), we have:

**(x² - y²) / (x² + y²) = cos²θ - sin²θ**

This expression is equivalent to **cos(2θ)**, demonstrating the connection between the original expression and trigonometric functions.

### Applications and Further Exploration

The expression (x² - y²) / (x² + y²) finds applications in various fields:

**Physics:**This expression can represent the ratio of kinetic energy to total energy in a certain physical system.**Geometry:**It can be used to derive various geometric relationships.**Calculus:**The expression can appear in derivatives and integrals.

Further exploration of this expression can involve:

**Finding its maximum and minimum values.****Graphing the expression in different coordinate systems.****Analyzing its behavior as x and y approach infinity.**

By exploring the expression (x² - y²) / (x² + y²) through factorization, analysis, and connections to other mathematical concepts, we gain deeper insights into its properties and its role in various fields.