The expression
\[\frac{P+Q}{P-Q}-\frac{P-Q}{P+Q}\]
where $P=x+y$ and $Q=x-y$, is equivalent to:
$\textbf{(A)}\ \frac{x^2-y^2}{xy}\qquad \textbf{(B)}\ \frac{x^2-y^2}{2xy}\qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ \frac{x^2+y^2}{xy}\qquad \textbf{(E)}\ \frac{x^2+y^2}{2xy}$
Simplifying before substituting by obtaining a common denominator gives: 
$\frac{(P+Q)^2 - (P - Q)^2}{(P-Q)(P+Q)}$
$\frac{(P^2 + 2PQ + Q^2) - (P^2 - 2PQ + Q^2)}{P^2 - Q^2}$
$\frac{4PQ}{P^2 - Q^2}$
Substituting gives:
$\frac{4(x+y)(x-y)}{(x+y)^2 - (x - y)^2}$
$\frac{4(x^2 - y^2)}{(x^2 + 2xy + y^2) - (x^2 - 2xy + y^2)}$
$\frac{4(x^2 - y^2)}{4xy}$
$\frac{x^2 - y^2}{xy}$
$\boxed{\textbf{(A)}}$