4 min read Jun 17, 2024

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.

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