How can I simplify this expression using basic algebra?

[trixitium]

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.

Homework Statement

Simplify:

Homework Equations

$\left(x + \frac{1}{x} \right)\left(y + \frac{1}{y} \right) + \left(x - \frac{1}{x} \right)\left(y - \frac{1}{y} \right)$

The Attempt at a Solution

$\left(x + \frac{1}{x} \right)\left(y + \frac{1}{y} \right) + \left(x - \frac{1}{x} \right)\left(y - \frac{1}{y} \right) \ =$


$\left(\frac{x^2 + 1}{x} \right)\left(\frac{y^2 + 1}{y} \right)+ \left(\frac{x^2 - 1}{x} \right)\left(\frac{y^2 - 1}{y} \right) \ =$

$\frac{1}{xy} \left( \left(x^2+1 \right) \left(y^2 + 1\right)+ \left(x^2 - 1\right) \left(y^2 - 1 \right) \right) \ = \ ... \ =$

$\frac{2(x^2y^2 + 1)}{xy}$

Thanks!

 

[jbunniii]

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

$$\left(x + \frac{1}{x}\right) \left(y + \frac{1}{y}\right) + \left(x - \frac{1}{x}\right) \left(y - \frac{1}{y}\right) =$$

$$\left(xy + \frac{x}{y} + \frac{y}{x} + \frac{1}{xy}\right) + \left(xy - \frac{x}{y} - \frac{y}{x} + \frac{1}{xy}\right) =$$

$$\left(xy + \frac{1}{xy}\right) + \left(xy + \frac{1}{xy}\right) = \ldots$$