We are tasked with simplifying the expression:

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

Let's simplify it step by step.

Step 1: Common Denominator

To combine the two terms, let's write them as a single fraction using a common denominator: $\frac{\sqrt{3 + \sqrt{5}}}{\sqrt{3 - \sqrt{5}}} + \frac{\sqrt{3 - \sqrt{5}}}{\sqrt{3 + \sqrt{5}}} = \frac{(\sqrt{3 + \sqrt{5}})^2 + (\sqrt{3 - \sqrt{5}})^2}{\sqrt{(3 - \sqrt{5})(3 + \sqrt{5})}}$

Step 2: Simplify the Denominator

The denominator simplifies using the difference of squares formula: $(3 + \sqrt{5})(3 - \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4$

So, the denominator becomes: $\frac{(\sqrt{3 + \sqrt{5}})^2 + (\sqrt{3 - \sqrt{5}})^2}{\sqrt{4}} = \frac{(\sqrt{3 + \sqrt{5}})^2 + (\sqrt{3 - \sqrt{5}})^2}{2}$

Step 3: Simplify the Numerator

Now, let's focus on simplifying the numerator. Since $(\sqrt{a})^2 = a$, we get: $(\sqrt{3 + \sqrt{5}})^2 + (\sqrt{3 - \sqrt{5}})^2 = (3 + \sqrt{5}) + (3 - \sqrt{5}) = 3 + \sqrt{5} + 3 - \sqrt{5} = 6$

Step 4: Final Expression

Substituting back, we get: $\frac{6}{2} = 3$

Thus, the simplified value of the given expression is:

$\boxed{3}$

Would you like more details on any part of this solution? Here are some related questions to expand your understanding:

1. How can the difference of squares formula be used in complex expressions?
2. What are other cases where combining fractions helps simplify radical expressions?
3. Can you derive similar simplifications for expressions with cube roots?
4. How do we simplify nested radicals like $\sqrt{3 + \sqrt{5}}$?
5. What are some general strategies for simplifying radical expressions?

Tip: When working with radicals, it's often helpful to identify conjugates and apply the difference of squares formula for simplifications.