If `sin^(4) alpha + 4 cos^(4) beta + 2 = 4sqrt(2) sin alpha cos beta, alpha beta in [0, pi]`, then `cos (alpha + beta) - cos (alpha - beta)` is equal to `-sqrt(k)`. The value of k is _________.

Explanation

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

\[ \sin^4 \alpha + 4 \cos^4 \beta + 2 = 4\sqrt{2} \sin \alpha \cos \beta \]

### Step 1: Rearranging the Equation
We can rearrange the equation by moving all terms to one side:

\[ \sin^4 \alpha + 4 \cos^4 \beta - 4\sqrt{2} \sin \alpha \cos \beta + 2 = 0 \]

### Step 2: Dividing by 4
Next, we divide the entire equation by 4:

\[ \frac{\sin^4 \alpha}{4} + \cos^4 \beta - \sqrt{2} \sin \alpha \cos \beta + \frac{1}{2} = 0 \]

### Step 3: Applying AM-GM Inequality
Using the AM-GM inequality, we can express the left-hand side in a more manageable form. Let:

\[ x = \sin^4 \alpha, \quad y = \cos^4 \beta \]

Then we have:

\[ \frac{x}{4} + y + \frac{1}{2} \geq 3\sqrt[3]{\frac{xy}{4}} \quad \text{(AM-GM)} \]

### Step 4: Finding Equality Condition
For equality in AM-GM, we need:

\[ \frac{x}{4} = y = \frac{1}{2} \]

This implies:

1. \( \sin^4 \alpha = 2 \)
2. \( \cos^4 \beta = \frac{1}{2} \)

### Step 5: Solving for \(\alpha\) and \(\beta\)
From \( \sin^4 \alpha = 2 \), we have:

\[ \sin \alpha = 1 \implies \alpha = \frac{\pi}{2} \]

From \( \cos^4 \beta = \frac{1}{2} \), we have:

\[ \cos \beta = \frac{1}{\sqrt{2}} \implies \beta = \frac{\pi}{4} \]

### Step 6: Finding \( \cos(\alpha + \beta) - \cos(\alpha - \beta) \)
Now we need to calculate:

\[ \cos(\alpha + \beta) - \cos(\alpha - \beta) \]

Substituting the values of \(\alpha\) and \(\beta\):

\[ \cos\left(\frac{\pi}{2} + \frac{\pi}{4}\right) - \cos\left(\frac{\pi}{2} - \frac{\pi}{4}\right) \]

Calculating each term:

1. \( \cos\left(\frac{3\pi}{4}\right) = -\frac{1}{\sqrt{2}} \)
2. \( \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \)

Thus,

\[ -\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2} \]

### Step 7: Relating to \(-\sqrt{k}\)
We have:

\[ \cos(\alpha + \beta) - \cos(\alpha - \beta) = -\sqrt{2} \]

This implies:

\[ -\sqrt{k} = -\sqrt{2} \implies k = 2 \]

### Final Answer
The value of \( k \) is:

\[ \boxed{2} \]