An equivalent of the expression &((x^2+1)/(x)) ((y^2+1)/(y)) & + ((x^2-1)/(y)) ((y^2-1)/(x)), & xy≠0, is:

1967 AMC 12 Problem 2

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An equivalent of the expression

\begin{aligned} &\left(\frac{x^2+1}{x}\right) \left(\frac{y^2+1}{y}\right)\\ &\quad+ \left(\frac{x^2-1}{y}\right) \left(\frac{y^2-1}{x}\right),\\ &\qquad xy\ne0, \end{aligned}

is:

1

2xy

2x^2y^2+2

2xy+\dfrac{2}{xy}

\dfrac{2x}{y}+\dfrac{2y}{x}

Expand each product into four terms

Look for cancellation of the

\frac{x}{y}

and

\frac{y}{x}

terms

The first product is

xy+\frac{x}{y}+\frac{y}{x}+\frac1{xy},

while the second is

xy-\frac{x}{y}-\frac{y}{x}+\frac1{xy}.

Adding gives

2xy+\frac{2}{xy}.

Therefore, the correct answer is D.