Find the value of `(cos^(2)66^(@)-sin^(2)6^(@))(cos^(2)48^(@)-sin^(2)12^(@))`.

Explanation

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

\[ (cos^2 66^\circ - sin^2 6^\circ)(cos^2 48^\circ - sin^2 12^\circ) \]

### Step 1: Use the identity \( \cos^2 A - \sin^2 B = \cos(A + B) \cos(A - B) \)

For the first part \( \cos^2 66^\circ - \sin^2 6^\circ \):

\[ \cos^2 66^\circ - \sin^2 6^\circ = \cos(66^\circ + 6^\circ) \cos(66^\circ - 6^\circ) \]

Calculating the angles:

\[ \cos(66^\circ + 6^\circ) = \cos 72^\circ \]
\[ \cos(66^\circ - 6^\circ) = \cos 60^\circ \]

Thus,

\[ \cos^2 66^\circ - \sin^2 6^\circ = \cos 72^\circ \cos 60^\circ \]

### Step 2: Calculate \( \cos 60^\circ \)

We know:

\[ \cos 60^\circ = \frac{1}{2} \]

So,

\[ \cos^2 66^\circ - \sin^2 6^\circ = \cos 72^\circ \cdot \frac{1}{2} \]

### Step 3: For the second part \( \cos^2 48^\circ - \sin^2 12^\circ \)

Using the same identity:

\[ \cos^2 48^\circ - \sin^2 12^\circ = \cos(48^\circ + 12^\circ) \cos(48^\circ - 12^\circ) \]

Calculating the angles:

\[ \cos(48^\circ + 12^\circ) = \cos 60^\circ \]
\[ \cos(48^\circ - 12^\circ) = \cos 36^\circ \]

Thus,

\[ \cos^2 48^\circ - \sin^2 12^\circ = \cos 60^\circ \cos 36^\circ \]

### Step 4: Substitute back into the expression

Now substituting both parts back into the original expression:

\[ (cos^2 66^\circ - sin^2 6^\circ)(cos^2 48^\circ - sin^2 12^\circ) = \left(\cos 72^\circ \cdot \frac{1}{2}\right) \left(\cos 60^\circ \cdot \cos 36^\circ\right) \]

### Step 5: Substitute the known values

We already know:

\[ \cos 60^\circ = \frac{1}{2} \]

So we have:

\[ = \left(\cos 72^\circ \cdot \frac{1}{2}\right) \left(\frac{1}{2} \cdot \cos 36^\circ\right) \]
\[ = \frac{1}{4} \cos 72^\circ \cos 36^\circ \]

### Step 6: Use the identity \( \cos 72^\circ = \sin 18^\circ \)

We know:

\[ \cos 72^\circ = \sin 18^\circ \]

Thus:

\[ = \frac{1}{4} \sin 18^\circ \cos 36^\circ \]

### Step 7: Calculate \( \sin 18^\circ \) and \( \cos 36^\circ \)

Using known values:

\[ \sin 18^\circ = \frac{\sqrt{5}-1}{4}, \quad \cos 36^\circ = \frac{\sqrt{5}+1}{4} \]

### Step 8: Final Calculation

Substituting these values:

\[ = \frac{1}{4} \cdot \frac{\sqrt{5}-1}{4} \cdot \frac{\sqrt{5}+1}{4} \]
\[ = \frac{(\sqrt{5}-1)(\sqrt{5}+1)}{64} \]
\[ = \frac{5-1}{64} = \frac{4}{64} = \frac{1}{16} \]

### Final Answer

Thus, the value of the expression is:

\[ \frac{1}{16} \]