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 problem, we start with the equation given:

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

### Step 1: Rearranging the Equation
We can rearrange the equation to isolate the terms on one side:

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

### Step 2: Applying AM-GM Inequality
We can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality to the terms \(\sin^4 \alpha\), \(4 \cos^4 \beta\), and \(2\).

According to AM-GM, we have:

\[ \frac{\sin^4 \alpha + 4 \cos^4 \beta + 2}{3} \geq \sqrt[3]{\sin^4 \alpha \cdot (4 \cos^4 \beta) \cdot 2} \]

### Step 3: Finding Minimum Values
To find the minimum values, we set:

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

Then, we can rewrite the equation as:

\[ x + 4y + 2 = 4\sqrt{2} \sqrt[4]{xy} \]

### Step 4: Setting Equal Values
From AM-GM, equality holds when all terms are equal:

\[ \sin^4 \alpha = 4 \cos^4 \beta = 1 \]

This implies:

\[ \sin^4 \alpha = 1 \quad \Rightarrow \quad \sin \alpha = 1 \quad \Rightarrow \quad \alpha = \frac{\pi}{2} \]

And for \(\cos^4 \beta\):

\[ 4 \cos^4 \beta = 1 \quad \Rightarrow \quad \cos^4 \beta = \frac{1}{4} \quad \Rightarrow \quad \cos \beta = \frac{1}{\sqrt{2}} \quad \Rightarrow \quad \beta = \frac{\pi}{4} \]

### Step 5: 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:

\[ \cos\left(\frac{3\pi}{4}\right) = -\frac{1}{\sqrt{2}}, \quad \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

Thus, we have:

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

### Step 6: Relating to the Given Expression
According to the problem, we have:

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

From our calculation, we have:

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

### Step 7: Solving for \(k\)
Thus, we find:

\[ k = 2 \]

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

\[ \boxed{2} \]