If x `=(sqrt(5)-2)/(sqrt(5)+2)` and `y = (sqrt(5)+2)/(sqrt(5)-2)` : find :
xy

Answer

To find the product \( xy \) where \( x = \frac{\sqrt{5} - 2}{\sqrt{5} + 2} \) and \( y = \frac{\sqrt{5} + 2}{\sqrt{5} - 2} \), we can follow these steps:

### Step 1: Write down the expression for \( xy \)

We have:
\[ xy = \left(\frac{\sqrt{5} - 2}{\sqrt{5} + 2}\right) \left(\frac{\sqrt{5} + 2}{\sqrt{5} - 2}\right) \]

### Step 2: Simplify the expression

When we multiply the two fractions, we can see that the numerator of the first fraction and the denominator of the second fraction will cancel out:
\[ xy = \frac{(\sqrt{5} - 2)(\sqrt{5} + 2)}{(\sqrt{5} + 2)(\sqrt{5} - 2)} \]

### Step 3: Apply the difference of squares formula

The expression can be simplified further using the difference of squares formula \( a^2 - b^2 = (a - b)(a + b) \):
\[ xy = \frac{(\sqrt{5})^2 - (2)^2}{(\sqrt{5})^2 - (2)^2} \]

### Step 4: Calculate the squares

Calculating the squares gives us:
\[ xy = \frac{5 - 4}{5 - 4} \]

### Step 5: Simplify the fraction

Now, simplifying the fraction:
\[ xy = \frac{1}{1} = 1 \]

### Final Answer

Thus, the product \( xy \) is:
\[ \boxed{1} \]