If `(2+sqrt(5))/(2-sqrt(5))` =x and `(2-sqrt(5))/(2+sqrt(5))` =y , find the value of `x^(2)-y^(2)`.

Answer

To solve the problem, we need to find the value of \( x^2 - y^2 \) given:

\[ x = \frac{2 + \sqrt{5}}{2 - \sqrt{5}} \quad \text{and} \quad y = \frac{2 - \sqrt{5}}{2 + \sqrt{5}} \]

### Step 1: Calculate \( x^2 \)

First, we will calculate \( x^2 \):

\[ x^2 = \left( \frac{2 + \sqrt{5}}{2 - \sqrt{5}} \right)^2 = \frac{(2 + \sqrt{5})^2}{(2 - \sqrt{5})^2} \]

Using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \) for the numerator and \( (a - b)^2 = a^2 - 2ab + b^2 \) for the denominator:

- Numerator:
\[ (2 + \sqrt{5})^2 = 2^2 + 2 \cdot 2 \cdot \sqrt{5} + (\sqrt{5})^2 = 4 + 4\sqrt{5} + 5 = 9 + 4\sqrt{5} \]

- Denominator:
\[ (2 - \sqrt{5})^2 = 2^2 - 2 \cdot 2 \cdot \sqrt{5} + (\sqrt{5})^2 = 4 - 4\sqrt{5} + 5 = 9 - 4\sqrt{5} \]

Thus, we have:

\[ x^2 = \frac{9 + 4\sqrt{5}}{9 - 4\sqrt{5}} \]

### Step 2: Calculate \( y^2 \)

Next, we calculate \( y^2 \):

\[ y^2 = \left( \frac{2 - \sqrt{5}}{2 + \sqrt{5}} \right)^2 = \frac{(2 - \sqrt{5})^2}{(2 + \sqrt{5})^2} \]

Using the same formulas as above:

- Numerator:
\[ (2 - \sqrt{5})^2 = 4 - 4\sqrt{5} + 5 = 9 - 4\sqrt{5} \]

- Denominator:
\[ (2 + \sqrt{5})^2 = 4 + 4\sqrt{5} + 5 = 9 + 4\sqrt{5} \]

Thus, we have:

\[ y^2 = \frac{9 - 4\sqrt{5}}{9 + 4\sqrt{5}} \]

### Step 3: Calculate \( x^2 - y^2 \)

Now, we need to find \( x^2 - y^2 \):

\[ x^2 - y^2 = \frac{9 + 4\sqrt{5}}{9 - 4\sqrt{5}} - \frac{9 - 4\sqrt{5}}{9 + 4\sqrt{5}} \]

To subtract these fractions, we need a common denominator:

\[ x^2 - y^2 = \frac{(9 + 4\sqrt{5})^2 - (9 - 4\sqrt{5})^2}{(9 - 4\sqrt{5})(9 + 4\sqrt{5})} \]

Using the difference of squares formula \( a^2 - b^2 = (a - b)(a + b) \):

- Let \( a = 9 + 4\sqrt{5} \) and \( b = 9 - 4\sqrt{5} \):

\[ a - b = (9 + 4\sqrt{5}) - (9 - 4\sqrt{5}) = 8\sqrt{5} \]
\[ a + b = (9 + 4\sqrt{5}) + (9 - 4\sqrt{5}) = 18 \]

Thus, we have:

\[ (9 + 4\sqrt{5})^2 - (9 - 4\sqrt{5})^2 = (8\sqrt{5})(18) = 144\sqrt{5} \]

Now, we need to calculate the denominator:

\[ (9 - 4\sqrt{5})(9 + 4\sqrt{5}) = 9^2 - (4\sqrt{5})^2 = 81 - 80 = 1 \]

So, we have:

\[ x^2 - y^2 = \frac{144\sqrt{5}}{1} = 144\sqrt{5} \]

### Final Answer

Thus, the value of \( x^2 - y^2 \) is:

\[ \boxed{144\sqrt{5}} \]