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Simplify for x=1/y

Source: 1978 AHSME Problem 3

May 27, 2014
AMC

Problem Statement

For all non-zero numbers xx and yy such that x=1/yx = 1/y, (x−1x)(y+1y)\left(x-\frac{1}{x}\right)\left(y+\frac{1}{y}\right) equals
<spanclass=′latex−bold′>(A)</span>2x2<spanclass=′latex−bold′>(B)</span>2y2<spanclass=′latex−bold′>(C)</span>x2+y2<spanclass=′latex−bold′>(D)</span>x2−y2<spanclass=′latex−bold′>(E)</span>y2−x2<span class='latex-bold'>(A) </span>2x^2\qquad<span class='latex-bold'>(B) </span>2y^2\qquad<span class='latex-bold'>(C) </span>x^2+y^2\qquad<span class='latex-bold'>(D) </span>x^2-y^2\qquad <span class='latex-bold'>(E) </span>y^2-x^2
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