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Exercise on simplification

  1. Aug 7, 2012 #1
    Hello,

    I would like to solve this exercise in the best way as possible. I solved using the most trivial way and I am in doubt if are there some better way to solve.

    1. The problem statement, all variables and given/known data

    Simplify:

    2. Relevant equations

    [itex]\left(x + \frac{1}{x} \right)\left(y + \frac{1}{y} \right) + \left(x - \frac{1}{x} \right)\left(y - \frac{1}{y} \right)[/itex]

    3. The attempt at a solution

    [itex]\left(x + \frac{1}{x} \right)\left(y + \frac{1}{y} \right) + \left(x - \frac{1}{x} \right)\left(y - \frac{1}{y} \right) \ = [/itex]


    [itex] \left(\frac{x^2 + 1}{x} \right)\left(\frac{y^2 + 1}{y} \right)+ \left(\frac{x^2 - 1}{x} \right)\left(\frac{y^2 - 1}{y} \right) \ = [/itex]

    [itex] \frac{1}{xy} \left( \left(x^2+1 \right) \left(y^2 + 1\right)+ \left(x^2 - 1\right) \left(y^2 - 1 \right) \right) \ = \ ... \ = [/itex]

    [itex] \frac{2(x^2y^2 + 1)}{xy}[/itex]

    Thanks!
     
  2. jcsd
  3. Aug 7, 2012 #2

    jbunniii

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    Science Advisor
    Homework Helper
    Gold Member

    I don't see anything wrong with what you have done. A different, but not necessarily any better or quicker, way to do it is

    [tex]\left(x + \frac{1}{x}\right) \left(y + \frac{1}{y}\right) + \left(x - \frac{1}{x}\right) \left(y - \frac{1}{y}\right) =[/tex]

    [tex]\left(xy + \frac{x}{y} + \frac{y}{x} + \frac{1}{xy}\right) + \left(xy - \frac{x}{y} - \frac{y}{x} + \frac{1}{xy}\right) = [/tex]

    [tex]\left(xy + \frac{1}{xy}\right) + \left(xy + \frac{1}{xy}\right) = \ldots[/tex]
     
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