Brahmagupta-Fibonacci Identity/General/Corollary

Corollary to General Version of Brahmagupta-Fibonacci Identity

Let $a, b, c, d, n$ be numbers.

Then:

$\paren {a^2 + n b^2} \paren {c^2 + n d^2} = \paren {a c - n b d}^2 + n \paren {a d + b c}^2$

$\ds \paren {a^2 + n b^2} \paren {c^2 + n d^2}$

$=$

$\ds \paren {a c + n b d}^2 + n \paren {a d - b c}^2$

$\ds \leadsto \ \ $

$\ds \paren {a^2 + n \paren {-b}^2} \paren {c^2 + n d^2}$

$=$

$\ds \paren {a c + n \paren {-b} d}^2 + n \paren {a d - \paren {-b} c}^2$

substituting $-b$ for $b$

$\ds \leadsto \ \ $

$\ds \paren {a^2 + n b^2} \paren {c^2 + n d^2}$

$=$

$\ds \paren {a c - n b d}^2 + n \paren {a d + b c}^2$

$\blacksquare$