Find the value of `4cos[cos^(-1)((1)/(4)(sqrt(6)-sqrt(2)))-cos^(-1)((1)/(4)(sqrt(6)+sqrt(2)))]`.

Answer

To solve the problem, we need to find the value of

\[ 4 \cos \left[ \cos^{-1} \left( \frac{1}{4}(\sqrt{6} - \sqrt{2}) \right) - \cos^{-1} \left( \frac{1}{4}(\sqrt{6} + \sqrt{2}) \right) \right]. \]

### Step 1: Use the identity for the difference of cosines
We can use the identity:

\[ \cos^{-1}(x) - \cos^{-1}(y) = \cos^{-1}(xy + \sqrt{(1-x^2)(1-y^2)}) \]

Let \( x = \frac{1}{4}(\sqrt{6} - \sqrt{2}) \) and \( y = \frac{1}{4}(\sqrt{6} + \sqrt{2}) \).

### Step 2: Calculate \( xy \)
First, we calculate \( xy \):

\[ xy = \left(\frac{1}{4}(\sqrt{6} - \sqrt{2})\right) \left(\frac{1}{4}(\sqrt{6} + \sqrt{2})\right) = \frac{1}{16} \left((\sqrt{6})^2 - (\sqrt{2})^2\right) = \frac{1}{16}(6 - 2) = \frac{4}{16} = \frac{1}{4}. \]

### Step 3: Calculate \( 1 - x^2 \) and \( 1 - y^2 \)
Next, we calculate \( 1 - x^2 \) and \( 1 - y^2 \):

\[ x^2 = \left(\frac{1}{4}(\sqrt{6} - \sqrt{2})\right)^2 = \frac{1}{16}(6 - 2\sqrt{12} + 2) = \frac{1}{16}(8 - 2\sqrt{12}) = \frac{1}{16}(8 - 4\sqrt{3}), \]

\[ 1 - x^2 = 1 - \frac{8 - 4\sqrt{3}}{16} = \frac{16 - (8 - 4\sqrt{3})}{16} = \frac{8 + 4\sqrt{3}}{16} = \frac{2 + \sqrt{3}}{4}. \]

Similarly for \( y \):

\[ y^2 = \left(\frac{1}{4}(\sqrt{6} + \sqrt{2})\right)^2 = \frac{1}{16}(6 + 2\sqrt{12} + 2) = \frac{1}{16}(8 + 4\sqrt{3}), \]

\[ 1 - y^2 = 1 - \frac{8 + 4\sqrt{3}}{16} = \frac{16 - (8 + 4\sqrt{3})}{16} = \frac{8 - 4\sqrt{3}}{16} = \frac{2 - \sqrt{3}}{4}. \]

### Step 4: Calculate \( \sqrt{(1-x^2)(1-y^2)} \)
Now we calculate \( \sqrt{(1-x^2)(1-y^2)} \):

\[ (1-x^2)(1-y^2) = \left(\frac{2 + \sqrt{3}}{4}\right) \left(\frac{2 - \sqrt{3}}{4}\right) = \frac{(2+\sqrt{3})(2-\sqrt{3})}{16} = \frac{4 - 3}{16} = \frac{1}{16}. \]

Thus,

\[ \sqrt{(1-x^2)(1-y^2)} = \frac{1}{4}. \]

### Step 5: Combine results
Now we combine the results:

\[ \cos^{-1} \left( \frac{1}{4} \right) + \frac{1}{4} = \cos^{-1} \left( \frac{1}{4} + \frac{1}{4} \right) = \cos^{-1} \left( \frac{1}{2} \right) = \frac{\pi}{3}. \]

### Step 6: Final calculation
Now we substitute back into the original expression:

\[ 4 \cos \left( \frac{\pi}{3} \right) = 4 \cdot \frac{1}{2} = 2. \]

Thus, the final answer is:

\[ \boxed{2}. \]