What Is the Value of a+b If (a+√(a²+1))(b+√(b²+1))=1?

[anemone]

Here is this week's POTW:

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Suppose that for two real numbers $a$ and $b$ the following equality is true:

$(a+\sqrt{a^2+1})(b+\sqrt{b^2+1})=1$.

Find the value of $a+b$.

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[anemone]

Congratulations to the following members for their correct solution::)
1. kaliprasad
2. lfdahl
3. Theia

Solution from Theia:

Spoiler

If the product of two numbers is equal to one, they have to be reciprocals. So if

\(\displaystyle a + \sqrt{a^2 +1} = u \qquad \Rightarrow \qquad a = \frac{u^2 - 1}{2u}\)

then

\(\displaystyle b + \sqrt{b^2 +1} = \frac{1}{u} \qquad \Rightarrow \qquad b = \frac{u^{-2} - 1}{2u^{-1}}\).

Now, by direct simplification one obtains

\(\displaystyle a + b = \frac{u^2 - 1}{2u} + \frac{u}{2} \cdot \frac{1 - u^2}{u^2} = \frac{u^2 - 1}{2u} + \frac{1 - u^2}{2u} = 0\).