1. **State the problem:** Simplify the expression $$\frac{\sqrt{a+1} + \sqrt{a-1}}{\sqrt{a+1} - \sqrt{a-1}} - \frac{\sqrt{a+1} - \sqrt{a-1}}{\sqrt{a+1} + \sqrt{a-1}} \div \frac{\sqrt{a+1}}{a}$$ where $a > 1$ to ensure the square roots are defined. 2. **Rewrite the expression:** The expression is $$\left(\frac{\sqrt{a+1} + \sqrt{a-1}}{\sqrt{a+1} - \sqrt{a-1}} - \frac{\sqrt{a+1} - \sqrt{a-1}}{\sqrt{a+1} + \sqrt{a-1}}\right) \div \frac{\sqrt{a+1}}{a}$$ 3. **Simplify the subtraction inside the parentheses:** Find a common denominator: $$\frac{(\sqrt{a+1} + \sqrt{a-1})^2 - (\sqrt{a+1} - \sqrt{a-1})^2}{(\sqrt{a+1} - \sqrt{a-1})(\sqrt{a+1} + \sqrt{a-1})}$$ 4. **Use the difference of squares formula:** $$(x+y)^2 - (x-y)^2 = 4xy$$ Here, $x=\sqrt{a+1}$ and $y=\sqrt{a-1}$, so numerator becomes: $$4 \sqrt{a+1} \sqrt{a-1} = 4 \sqrt{(a+1)(a-1)} = 4 \sqrt{a^2 - 1}$$ 5. **Simplify the denominator:** $$(\sqrt{a+1} - \sqrt{a-1})(\sqrt{a+1} + \sqrt{a-1}) = (\sqrt{a+1})^2 - (\sqrt{a-1})^2 = (a+1) - (a-1) = 2$$ 6. **So the expression inside parentheses simplifies to:** $$\frac{4 \sqrt{a^2 - 1}}{2} = 2 \sqrt{a^2 - 1}$$ 7. **Now divide by $\frac{\sqrt{a+1}}{a}$:** $$\frac{2 \sqrt{a^2 - 1}}{1} \times \frac{a}{\sqrt{a+1}} = 2a \frac{\sqrt{a^2 - 1}}{\sqrt{a+1}}$$ 8. **Simplify $\sqrt{a^2 - 1}$:** $$\sqrt{a^2 - 1} = \sqrt{(a-1)(a+1)} = \sqrt{a-1} \sqrt{a+1}$$ 9. **Substitute back:** $$2a \frac{\sqrt{a-1} \sqrt{a+1}}{\sqrt{a+1}} = 2a \sqrt{a-1}$$ **Final answer:** $$\boxed{2a \sqrt{a-1}}$$