Question

((x/y)-(y/x))((x/y)+(y/x)-2)^-1

Answer 1

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

Answer 2

To simplify the expression ((x/y)-(y/x))((x/y)+(y/x)-2)^-1, we can begin by simplifying the expression within the parentheses ((x/y)+(y/x)-2).

Let's start by finding a common denominator for the terms (x/y) and (y/x). The common denominator is xy.

((x/y) + (y/x)) = ((x^2 + y^2)/(xy))

Now, let's subtract 2 from the expression:

((x^2 + y^2)/(xy)) - 2 = ((x^2 + y^2)/(xy)) - (2xy/xy)

To combine the terms, we have:

((x^2 + y^2 - 2xy)/(xy))

Now, let's substitute this simplified expression back into the original expression:

((x/y) - (y/x)) * ((x^2 + y^2 - 2xy)/(xy))^-1

To simplify further, we can rewrite the expression as:

((x/y) - (y/x)) / ((x^2 + y^2 - 2xy)/(xy))

To divide by a fraction, we can multiply by the reciprocal of that fraction:

((x/y) - (y/x)) * (xy/(x^2 + y^2 - 2xy))

Multiplying the numerators and denominators, we have:

((x^2y - y^2x)/(xy)) * (xy/(x^2 + y^2 - 2xy))

Simplifying further:

(x^2y - y^2x) / (x^2 + y^2 - 2xy)

This is the simplified form of the given expression: (x^2y - y^2x) / (x^2 + y^2 - 2xy).