1. The problem is to simplify the expression $$\frac{1}{\sqrt{n + \sqrt{n^2 - 1}}}$$. 2. Start by letting $$x = \sqrt{n + \sqrt{n^2 - 1}}$$, so the expression becomes $$\frac{1}{x}$$. 3. To simplify, consider the conjugate expression inside the square root. Note that $$\sqrt{n + \sqrt{n^2 - 1}}$$ and $$\sqrt{n - \sqrt{n^2 - 1}}$$ are conjugates. 4. Multiply numerator and denominator by $$\sqrt{n - \sqrt{n^2 - 1}}$$ to rationalize the denominator: $$\frac{1}{\sqrt{n + \sqrt{n^2 - 1}}} \times \frac{\sqrt{n - \sqrt{n^2 - 1}}}{\sqrt{n - \sqrt{n^2 - 1}}} = \frac{\sqrt{n - \sqrt{n^2 - 1}}}{\sqrt{(n + \sqrt{n^2 - 1})(n - \sqrt{n^2 - 1})}}$$ 5. Simplify the denominator using difference of squares: $$ (n + \sqrt{n^2 - 1})(n - \sqrt{n^2 - 1}) = n^2 - (n^2 - 1) = 1 $$ 6. So the expression simplifies to: $$ \sqrt{n - \sqrt{n^2 - 1}} $$ 7. Therefore, the simplified form of the original expression is: $$ \boxed{\sqrt{n - \sqrt{n^2 - 1}}} $$