The value of `((sqrt5+sqrt3)/(sqrt5-sqrt3))^2+((sqrt5-sqrt3)/(sqrt5+sqrt3))^2` is equal to

Options

A. 64
B. 62
C. 66
D. 68

Answer

To solve the expression \(\left(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}\right)^2 + \left(\frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}\right)^2\), we can follow these steps:

### Step 1: Let \( A = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \) and \( B = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} \)

We need to find \( A^2 + B^2 \).

### Step 2: Use the identity \( A^2 + B^2 = (A + B)^2 - 2AB \)

To use this identity, we first need to calculate \( A + B \) and \( AB \).

### Step 3: Calculate \( A + B \)

\[ A + B = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} \]

To add these fractions, we find a common denominator:

\[ A + B = \frac{(\sqrt{5} + \sqrt{3})^2 + (\sqrt{5} - \sqrt{3})^2}{(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})} \]

### Step 4: Calculate the numerator

Calculating the squares:

\[ (\sqrt{5} + \sqrt{3})^2 = 5 + 3 + 2\sqrt{15} = 8 + 2\sqrt{15} \]
\[ (\sqrt{5} - \sqrt{3})^2 = 5 + 3 - 2\sqrt{15} = 8 - 2\sqrt{15} \]

Adding these:

\[ A + B = \frac{(8 + 2\sqrt{15}) + (8 - 2\sqrt{15})}{(\sqrt{5})^2 - (\sqrt{3})^2} = \frac{16}{2} = 8 \]

### Step 5: Calculate \( AB \)

\[ AB = \left(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}\right) \left(\frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}\right) = \frac{(\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})}{(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})} \]

This simplifies to:

\[ AB = \frac{5 - 3}{2} = 1 \]

### Step 6: Substitute back into the identity

Now we substitute \( A + B \) and \( AB \) into the identity:

\[ A^2 + B^2 = (A + B)^2 - 2AB = 8^2 - 2 \cdot 1 = 64 - 2 = 62 \]

### Final Answer

Thus, the value of the expression is:

\[ \boxed{62} \]