1951 AHSME Problem 20

When simplified and expressed with negative exponents, the expression (x+y)^{-1}(x^{-1}+y^{-1}) is equal to:

\textbf{(A)}\ x^{-2}+2x^{-1}y^{-1}+y^{-2}\qquad\textbf{(B)}\ x^{-2}+2^{-1}x^{-1}y^{-1}+y^{-2}\qquad\textbf{(C)}\ x^{-1}y^{-1} \textbf{(D)}\ \text{some even number}\qquad\textbf{(E)}\ \text{zero}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.