The value of the expression `(1)/(4){(a+(1)/(a))^(2) - (a-(1)/(a))}` is :

Options

A. `(1)/(2)`
B. `(1)/(4)`
C. 1
D. 4

Answer

To solve the expression \(\frac{1}{4}\left\{(a + \frac{1}{a})^2 - (a - \frac{1}{a})\right\}\), we will follow these steps:

### Step 1: Expand \((a + \frac{1}{a})^2\)
Using the formula \((x + y)^2 = x^2 + 2xy + y^2\), we can expand:
\[ (a + \frac{1}{a})^2 = a^2 + 2 \cdot a \cdot \frac{1}{a} + \left(\frac{1}{a}\right)^2 = a^2 + 2 + \frac{1}{a^2} \]

### Step 2: Expand \((a - \frac{1}{a})\)
This expression is already simplified, so we leave it as:
\[ (a - \frac{1}{a}) = a - \frac{1}{a} \]

### Step 3: Substitute the expansions into the original expression
Now we substitute the expanded forms back into the expression:
\[ \frac{1}{4}\left\{(a^2 + 2 + \frac{1}{a^2}) - (a - \frac{1}{a})\right\} \]

### Step 4: Simplify the expression inside the brackets
Combine the terms:
\[ = a^2 + 2 + \frac{1}{a^2} - a + \frac{1}{a} \]
Rearranging gives:
\[ = a^2 - a + 2 + \frac{1}{a} + \frac{1}{a^2} \]

### Step 5: Factor the expression if possible
Notice that \(a^2 - a + 2 + \frac{1}{a} + \frac{1}{a^2}\) does not factor nicely, but we can still evaluate it.

### Step 6: Combine terms
Now, we take the entire expression and multiply by \(\frac{1}{4}\):
\[ = \frac{1}{4}(a^2 - a + 2 + \frac{1}{a} + \frac{1}{a^2}) \]

### Step 7: Evaluate for specific values of \(a\) (if necessary)
For simplicity, let's evaluate this expression at \(a = 1\):
\[ = \frac{1}{4}(1^2 - 1 + 2 + 1 + 1) = \frac{1}{4}(1) = \frac{1}{4} \]

### Conclusion
Thus, the value of the expression is \(\frac{1}{4}\).

### Final Answer
The correct option is \( \frac{1}{4} \).

---