Alg3.3.14

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If a=x+\sqrt{x^2+1}\,,then show that x=\frac{1}{2}(a-a^{-1})\,

Given a-x=\sqrt{x^2+1}\,

Squaring on both sides,we get

(a-x)^2=x^2+1\,

a^2+x^2-2ax=x^2+1\,

a^2-2ax=1,2ax=a^2-1,x=\frac{1}{2}(\frac{a^2-1}{a})\,

Simplifying

x=\frac{1}{2}(a-a^{-1})\,


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