If `2p=x+1/x` and `2q=y+1/y`, then the value of `((pq+sqrt((p^2-1)(q^2-1)))/(xy+1/(xy)))` is

Answer

To solve the problem, we start with the given equations:

1. \( 2p = x + \frac{1}{x} \)
2. \( 2q = y + \frac{1}{y} \)

We want to find the value of:

\[ \frac{pq + \sqrt{(p^2 - 1)(q^2 - 1)}}{xy + \frac{1}{xy}} \]

### Step 1: Find \( p^2 \)

Starting with the first equation, we square both sides:

\[ (2p)^2 = \left(x + \frac{1}{x}\right)^2 \]

This simplifies to:

\[ 4p^2 = x^2 + 2 + \frac{1}{x^2} \]

Rearranging gives:

\[ 4p^2 = x^2 + \frac{1}{x^2} + 2 \]

Thus,

\[ p^2 = \frac{x^2 + \frac{1}{x^2} + 2}{4} \]

### Step 2: Simplify \( p^2 - 1 \)

Now, we calculate \( p^2 - 1 \):

\[ p^2 - 1 = \frac{x^2 + \frac{1}{x^2} + 2}{4} - 1 = \frac{x^2 + \frac{1}{x^2} + 2 - 4}{4} = \frac{x^2 + \frac{1}{x^2} - 2}{4} \]

Notice that \( x^2 + \frac{1}{x^2} - 2 = \left(x - \frac{1}{x}\right)^2 \). Therefore,

\[ p^2 - 1 = \frac{\left(x - \frac{1}{x}\right)^2}{4} \]

### Step 3: Find \( q^2 \) and \( q^2 - 1 \)

Similarly, for \( q \):

\[ q^2 = \frac{y^2 + \frac{1}{y^2} + 2}{4} \]

Thus,

\[ q^2 - 1 = \frac{y^2 + \frac{1}{y^2} + 2 - 4}{4} = \frac{y^2 + \frac{1}{y^2} - 2}{4} = \frac{\left(y - \frac{1}{y}\right)^2}{4} \]

### Step 4: Calculate \( \sqrt{(p^2 - 1)(q^2 - 1)} \)

Now we can find \( \sqrt{(p^2 - 1)(q^2 - 1)} \):

\[ \sqrt{(p^2 - 1)(q^2 - 1)} = \sqrt{\left(\frac{\left(x - \frac{1}{x}\right)^2}{4}\right)\left(\frac{\left(y - \frac{1}{y}\right)^2}{4}\right)} = \frac{(x - \frac{1}{x})(y - \frac{1}{y})}{4} \]

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

Next, we find \( pq \):

\[ pq = \frac{(x + \frac{1}{x})(y + \frac{1}{y})}{4} = \frac{xy + x\frac{1}{y} + y\frac{1}{x} + \frac{1}{xy}}{4} \]

### Step 6: Substitute into the main expression

Now we substitute \( pq \) and \( \sqrt{(p^2 - 1)(q^2 - 1)} \) into the main expression:

\[ \frac{pq + \sqrt{(p^2 - 1)(q^2 - 1)}}{xy + \frac{1}{xy}} = \frac{\frac{xy + x\frac{1}{y} + y\frac{1}{x} + \frac{1}{xy}}{4} + \frac{(x - \frac{1}{x})(y - \frac{1}{y})}{4}}{xy + \frac{1}{xy}} \]

### Step 7: Simplify the numerator

The numerator becomes:

\[ \frac{xy + x\frac{1}{y} + y\frac{1}{x} + \frac{1}{xy} + (x - \frac{1}{x})(y - \frac{1}{y})}{4} \]

### Step 8: Simplify the denominator

The denominator is:

\[ xy + \frac{1}{xy} \]

### Step 9: Final simplification

After simplifying the entire expression, we find that it reduces to:

\[ \frac{1}{2} \]

Thus, the final answer is:

\[ \boxed{\frac{1}{2}} \]