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

Jasonbradley

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Jun 16, 2024

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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.

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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.