Evalutate : `(4-sqrt(5))/(4+sqrt(5))+ (4+sqrt(5))/(4-sqrt(5))`

Answer

To evaluate the expression \((4 - \sqrt{5})/(4 + \sqrt{5}) + (4 + \sqrt{5})/(4 - \sqrt{5})\), we can follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ \frac{4 - \sqrt{5}}{4 + \sqrt{5}} + \frac{4 + \sqrt{5}}{4 - \sqrt{5}} \]

### Step 2: Find a common denominator
The common denominator for the two fractions is \((4 + \sqrt{5})(4 - \sqrt{5})\).

### Step 3: Rewrite the expression with the common denominator
We can rewrite the expression as:
\[ \frac{(4 - \sqrt{5})^2 + (4 + \sqrt{5})^2}{(4 + \sqrt{5})(4 - \sqrt{5})} \]

### Step 4: Expand the numerators
Now we expand the numerators:
1. For \((4 - \sqrt{5})^2\):
\[ (4 - \sqrt{5})^2 = 4^2 - 2 \cdot 4 \cdot \sqrt{5} + (\sqrt{5})^2 = 16 - 8\sqrt{5} + 5 = 21 - 8\sqrt{5} \]
2. For \((4 + \sqrt{5})^2\):
\[ (4 + \sqrt{5})^2 = 4^2 + 2 \cdot 4 \cdot \sqrt{5} + (\sqrt{5})^2 = 16 + 8\sqrt{5} + 5 = 21 + 8\sqrt{5} \]

### Step 5: Combine the expanded numerators
Now we combine the two results:
\[ (21 - 8\sqrt{5}) + (21 + 8\sqrt{5}) = 21 + 21 = 42 \]

### Step 6: Simplify the denominator
Next, we simplify the denominator:
\[ (4 + \sqrt{5})(4 - \sqrt{5}) = 4^2 - (\sqrt{5})^2 = 16 - 5 = 11 \]

### Step 7: Write the final expression
Now we can write the final expression:
\[ \frac{42}{11} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{42}{11} \]