Question

If a=x+√x^2+1 then show that x=1/2(a-a^-3)​

Answer 1

Answer:

$a = x + \sqrt{x^2+1} \\\\a -x = \sqrt{x^2+1}\\\\(a-x)^2 = \sqrt{x^2+1}^2\\\\a^2+x^2 -2ax=x^2+1\\\\a^2-2ax=x^2+1-x^2\\\\a^2-2ax=1\\\\2ax=a^2-1\\\\x=\frac{a^2-1}{2a} \\\\x=\frac{1}{2}(\frac{a^2-1}{a} )\\\\x=\frac{1}{2} (\frac{a^2}{a}-\frac{1}{a}) \\\\x=\frac{1}{2}(a-a^-^1)$

Hence, proved.

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Answer 2

Do squaring on both sides, and leaving x on the LHS of the equation, bring everything else to the RHS.