## The Elegant Proof of a Fundamental Identity

The equation **(a² + b²)(c² + d²) = (ac + bd)² + (ad - bc)²** is a fundamental identity in mathematics, often used in various fields like algebra, geometry, and trigonometry. This equation holds true for all real numbers a, b, c, and d, making it a powerful tool for simplifying expressions and solving problems.

### A Visual Proof

One way to understand the equation is through a visual representation. Consider two squares with sides of lengths a + b and c + d.

- The area of the first square is
**(a + b)²**, which can be divided into four smaller rectangles: a², ab, ab, and b². - The area of the second square is
**(c + d)²**, similarly divided into c², cd, cd, and d².

!

Now, consider the product of these two areas: **(a + b)² (c + d)²**. This can be represented as the area of a larger rectangle with sides a + b and c + d. This rectangle can be divided into four smaller rectangles:

**ac**: The area of the first rectangle.**ad**: The area of the second rectangle.**bc**: The area of the third rectangle.**bd**: The area of the fourth rectangle.

The area of this larger rectangle can also be expressed as **(ac + bd)² + (ad - bc)²**.

Notice that the two expressions for the area of the larger rectangle are equivalent:

**(a + b)² (c + d)² = (ac + bd)² + (ad - bc)²**

Expanding the left side of the equation gives us:

**(a² + 2ab + b²)(c² + 2cd + d²) = (ac + bd)² + (ad - bc)²**

Further expanding and simplifying both sides of the equation leads to the same expression, proving the identity.

### Algebraic Proof

Alternatively, we can prove the identity algebraically by simply expanding both sides of the equation:

**(a² + b²)(c² + d²) = (ac + bd)² + (ad - bc)²**

Expanding the left side gives:

**a²c² + a²d² + b²c² + b²d²**

Expanding the right side gives:

**(a²c² + 2abcd + b²d²) + (a²d² - 2abcd + b²c²)**

Simplifying both sides results in:

**a²c² + a²d² + b²c² + b²d²**

Thus, both sides are equal, proving the identity.

### Applications

This identity has a wide range of applications in mathematics, including:

**Simplifying algebraic expressions:**This identity can be used to simplify complex algebraic expressions involving squares and products of sums.**Solving trigonometric equations:**This identity is often used to solve trigonometric equations, especially those involving sine and cosine functions.**Geometric problems:**This identity can be used to solve geometric problems involving areas, distances, and angles.

The elegant proof and diverse applications of the equation **(a² + b²)(c² + d²) = (ac + bd)² + (ad - bc)²** highlight its significance in mathematics and its usefulness in various fields.