The given expression is:

$\frac{\sqrt{3 + \sqrt{5}}}{\sqrt{3 - \sqrt{5}}} + \frac{\sqrt{3 - \sqrt{5}}}{\sqrt{3 + \sqrt{5}}}$

We will simplify this step by step.

Step 1: Combine the terms into a single fraction

Using the common denominator $\sqrt{3 - \sqrt{5}} \cdot \sqrt{3 + \sqrt{5}}$, we combine the terms into one fraction:

$\frac{\sqrt{3 + \sqrt{5}}}{\sqrt{3 - \sqrt{5}}} + \frac{\sqrt{3 - \sqrt{5}}}{\sqrt{3 + \sqrt{5}}} = \frac{(\sqrt{3 + \sqrt{5}})^2 + (\sqrt{3 - \sqrt{5}})^2}{\sqrt{(3 + \sqrt{5})(3 - \sqrt{5})}}$

Step 2: Simplify the denominator

Using the identity $(a + b)(a - b) = a^2 - b^2$, we simplify the denominator:

$(3 + \sqrt{5})(3 - \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4$

Thus, the denominator simplifies to $\sqrt{4} = 2$.

Step 3: Simplify the numerator

Now, we simplify the squares in the numerator:

$(\sqrt{3 + \sqrt{5}})^2 = 3 + \sqrt{5}$ $(\sqrt{3 - \sqrt{5}})^2 = 3 - \sqrt{5}$

So the numerator becomes:

$(3 + \sqrt{5}) + (3 - \sqrt{5}) = 3 + \sqrt{5} + 3 - \sqrt{5} = 6$

Step 4: Final expression

Thus, the entire expression becomes:

$\frac{6}{2} = 3$

Conclusion:

The value of the given expression is $\boxed{3}$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How would you handle more complicated radicals in both the numerator and denominator?
2. What happens if both terms under the square roots are negative?
3. How do we rationalize expressions with multiple square roots?
4. Can you apply a similar approach to cubic roots instead of square roots?
5. What is the significance of the identity $(a+b)(a-b) = a^2 - b^2$ in simplifying radicals?

Tip: When simplifying radicals, always check if you can apply conjugate pairs or special identities to reduce the complexity.