Problem: For all non-zero numbers xxx and yyy such that x=1y(x−1x)(y+1y)x=\dfrac{1}{y} \quad\left(x-\dfrac{1}{x}\right)\left(y+\dfrac{1}{y}\right)x=y1(x−x1)(y+y1) equals
Answer Choices:
A. 2x22 x^{2}2x2
B. 2y22 y^{2}2y2
C. x2+y2x^{2}+y^{2}x2+y2
D. x2−y2x^{2}-y^{2}x2−y2
E. y2−x2y^{2}-x^{2}y2−x2 Solution:
(x−1x)(y+1y)=(x−y)(x+y)=x2−y2\left(x-\dfrac{1}{x}\right)\left(y+\dfrac{1}{y}\right)=(x-y)(x+y)=x^{2}-y^{2}(x−x1)(y+y1)=(x−y)(x+y)=x2−y2.