Evaluate: `((a)/(2b) + (2b)/(a))^(2)- ((a)/(2b)- (2b)/(a))^(2)- 4`

Answer

To evaluate the expression \(\left(\frac{a}{2b} + \frac{2b}{a}\right)^{2} - \left(\frac{a}{2b} - \frac{2b}{a}\right)^{2} - 4\), we can follow these steps:

### Step 1: Define the terms
Let \(x = \frac{a}{2b}\) and \(y = \frac{2b}{a}\).

### Step 2: Rewrite the expression
The expression can be rewritten as:
\[ (x + y)^{2} - (x - y)^{2} - 4 \]

### Step 3: Apply the difference of squares
Using the identity \(A^{2} - B^{2} = (A - B)(A + B)\), we can simplify:
\[ (x + y)^{2} - (x - y)^{2} = [(x+y) - (x-y)][(x+y) + (x-y)] \]
This simplifies to:
\[ [2y][2x] = 4xy \]
So the expression becomes:
\[ 4xy - 4 \]

### Step 4: Substitute back for \(x\) and \(y\)
Now substitute back \(x\) and \(y\):
\[ xy = \left(\frac{a}{2b}\right)\left(\frac{2b}{a}\right) = 1 \]
Thus, we have:
\[ 4(1) - 4 = 4 - 4 = 0 \]

### Final Answer
The value of the expression is:
\[ \boxed{0} \]

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