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# Componendo and dividendo

Componendo and dividendo is a method of simplification in mathematics.

According to it,

$text\left\{if \right\} frac\left\{a\right\}\left\{b\right\} = frac\left\{c\right\}\left\{d\right\} text\left\{ then \right\} frac\left\{a+b\right\}\left\{a-b\right\} = frac\left\{c+d\right\}\left\{c-d\right\}.$

## Proof

Add one to both sides of the equality a / b = c / d to get

$frac\left\{a\right\}\left\{b\right\} + 1 = frac\left\{c\right\}\left\{d\right\} + 1 implies frac\left\{a+b\right\}\left\{b\right\} = frac\left\{c+d\right\}\left\{d\right\}. qquad \left(1\right)$
Now subtract one to get
$frac\left\{a\right\}\left\{b\right\} - 1 = frac\left\{c\right\}\left\{d\right\} - 1 implies frac\left\{a-b\right\}\left\{b\right\} = frac\left\{c-d\right\}\left\{d\right\}. qquad \left(2\right)$
The desired result follows after dividing (1) by (2).

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