Simplified value of `sqrt(17+12sqrt(2)) - tan ((pi)/(8))` is

Options

A. `0`
B. `1`
C. `-1`
D. `4+sqrt(2)`

Explanation

To simplify the expression \( \sqrt{17 + 12\sqrt{2}} - \tan\left(\frac{\pi}{8}\right) \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt{17 + 12\sqrt{2}} - \tan\left(\frac{\pi}{8}\right) \]

### Step 2: Identify the square root
We can express \( 17 \) and \( 12\sqrt{2} \) in a form that allows us to simplify the square root. Notice that:
\[ 17 = 3^2 + (2\sqrt{2})^2 \]
and
\[ 12\sqrt{2} = 2 \cdot 3 \cdot 2\sqrt{2} \]
This suggests that:
\[ \sqrt{17 + 12\sqrt{2}} = \sqrt{(3 + 2\sqrt{2})^2} \]

### Step 3: Simplify the square root
Taking the square root gives us:
\[ \sqrt{(3 + 2\sqrt{2})^2} = 3 + 2\sqrt{2} \]

### Step 4: Substitute back into the expression
Now we substitute back into our expression:
\[ 3 + 2\sqrt{2} - \tan\left(\frac{\pi}{8}\right) \]

### Step 5: Evaluate \( \tan\left(\frac{\pi}{8}\right) \)
The value of \( \tan\left(\frac{\pi}{8}\right) \) can be calculated using the half-angle formula:
\[ \tan\left(\frac{\pi}{8}\right) = \sqrt{2} - 1 \]

### Step 6: Substitute the value of \( \tan\left(\frac{\pi}{8}\right) \)
Substituting this value gives us:
\[ 3 + 2\sqrt{2} - (\sqrt{2} - 1) \]

### Step 7: Simplify the expression
Now we simplify:
\[ 3 + 2\sqrt{2} - \sqrt{2} + 1 = 3 + 1 + 2\sqrt{2} - \sqrt{2} = 4 + \sqrt{2} \]

### Final Answer
Thus, the simplified value of the expression is:
\[ \boxed{4 + \sqrt{2}} \]