Why is $\frac{\frac{a}{b} - \frac{b}{a}}{\frac{a+b}{ab}}$ = a - b? I'm given the complex rational expression
$$\frac{\frac{a}{b} - \frac{b}{a}}{\frac{a+b}{ab}}$$
and asked to simplify. The solution provided is $a - b$; however I get $\frac{a^2 - b^2}{a+b}$.
My working:
Numerator first: 
$$\frac{a}{b} - \frac{b}{a}$$
Least common denominator is $ab$.
Multiplying each part to get the LCD in the denominator on both sides I get:
$$\frac{a}{b} \cdot \frac{a}{a} - \frac{b}{a} \cdot \frac{b}{b}$$
$$= \frac{a^2 - b^2}{ab}$$
Multiplying this expression by the reciprocal in the original problem:
$$\frac{a^2 - b^2}{ab} \cdot \frac{ab}{a+b}$$
$$= \frac{ab(a^2-b^2)}{ab(a+b)}$$
Cancel out common factor $ab$:
$$\frac{a^2 - b^2}{a + b}$$
Where did I go wrong and how can I arrive at $a - b$?

Not that $$\frac{a}{b}-\frac{b}{a}=\frac{(a-b)(a+b)}{ab}$$ so we get
$$\frac{(a-b)(a+b)ab}{(a+b)ab}=a-b$$
