1. ## Simplifying an expression

Hi,

Does anyone know straightforward steps to make the following simplification:
$\displaystyle (x+\sqrt{x^2-1})^2-(x+\sqrt{x^2-1})^{-2}=4x\sqrt{x^2-1}$

I've tried to make the step:
$\displaystyle =\frac{(x+\sqrt{x^2-1})^4-1}{(x+\sqrt{x^2-1})^{2}}$

and multiplying out etc, with the aim of the simplified expression as a factor in the nominator, but then just get into a mess. I'd really like to understand how to spot when such a simplification would be possible.

Many thanks,

Chris

2. ## Re: Simplifying an expression

Originally Posted by entropyslave
Does anyone know straightforward steps to make the following simplification:
$\displaystyle (x+\sqrt{x^2-1})^2-(x+\sqrt{x^2-1})^{-2}=4x\sqrt{x^2-1}$
I have not done this, but if I were you I would factor the LHS as the sum&difference of two squares.

3. ## Re: Simplifying an expression

Thanks for your post. So I can do it now following your suggestion, however it seems very hard to see it without knowing the goal:

$\displaystyle =\frac{[(x+\sqrt{x^2-1})^2+1][(x+\sqrt{x^2-1})^2-1]}{(x+\sqrt{x^2-1})(x+\sqrt{x^2-1})}$

LHS of nominator is then straightforward:
$\displaystyle ={x^2+x\sqrt{x^2-1}+x^2+x\sqrt{x^2-1}=2x(x+\sqrt{x^2-1})}$

but the RHS presents some difficulty without the hindsight of the answer. Grouping the terms (with explicit surd factors) makes the factorisation clear:

$\displaystyle = {2[\sqrt{x^2-1}\sqrt{x^2-1}+x\sqrt{x^2-1}]=2\sqrt{x^2-1}(x+\sqrt{x^2-1})$

Now of course the denominator cancels and you're left with:

$\displaystyle = 4x\sqrt{x^2-1}$

Any tips for seeing this pattern in advance?

Thanks