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Math Help - Simplifying an expression

  1. #1
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    Simplifying an expression

    Hi,

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

    I've tried to make the step:
    =\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
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  2. #2
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    Re: Simplifying an expression

    Quote Originally Posted by entropyslave View Post
    Does anyone know straightforward steps to make the following simplification:
    (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.
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  3. #3
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    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:

    =\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:
    ={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:

    = {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:

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

    Any tips for seeing this pattern in advance?

    Thanks
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