If (4a + 5b)(4c - 5d) = (4a - 5b)(4c + 5d), prove that a, b, c, d are in proportion.

Given, (4a + 5b)(4c - 5d) = (4a - 5b)(4c + 5d).

On cross-multiplication,

$\Rightarrow \dfrac{4a + 5b}{4a - 5b} = \dfrac{4c + 5d}{4c - 5d} \\[0.5em]$

By componendo and dividendo,

$\Rightarrow \dfrac{4a + 5b + 4a - 5b}{4a + 5b - 4a + 5b} = \dfrac{4c + 5d + 4c - 5d}{4c + 5d - 4c + 5d} \\[0.5em] \Rightarrow \dfrac{8a}{10b} = \dfrac{8c}{10d} \\[0.5em]$

On dividing the equation by $\dfrac{8}{10}$,

$\Rightarrow \dfrac{a}{b} = \dfrac{c}{d} \\[0.5em] \Rightarrow a : b : : c : d.$

Hence, proved that a, b, c, d are in proportion.