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The method of Componendo et Dividendo allows a quick way to do some calculations, and can simplify the amount of expansion needed.
If and are numbers such that are non-zero and , then
This can be proven directly by observing that
1. Show the converse, namely that if and are numbers such that are non-zero and , then .
Solution: We apply Componendo et Dividendo with (which is valid since ), and get that
Note: The converse of Componendo and Dividendo also holds, and we can prove it by applying Dividendo and Componendo respectively.
2. Solve for : .
Solution: For the fractions to make sense, we must have .
Cross multiplying, we get
Apply Componendo et Dividendo with (which is valid since ), we get that . However since , we have as the only solution.
Note: We also need to check the condition that the denominators are non-zero, but this is obvious.