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

### Step-by-Step Solution:

1. **Apply the Trigonometric Identity**:
We use the identity \( \cos^2 A - \sin^2 B = \cos(A + B) \cos(A - B) \).
- For \( \cos^2 66^\circ - \sin^2 6^\circ \):
\[ \cos^2 66^\circ - \sin^2 6^\circ = \cos(66^\circ + 6^\circ) \cos(66^\circ - 6^\circ) = \cos 72^\circ \cos 60^\circ \]

2. **Calculate \( \cos 72^\circ \) and \( \cos 60^\circ \)**:
- We know that \( \cos 60^\circ = \frac{1}{2} \).
- \( \cos 72^\circ \) can be calculated using the known value or derived from \( \sin 18^\circ \) since \( \cos 72^\circ = \sin 18^\circ \).

3. **Calculate \( \cos^2 48^\circ - \sin^2 12^\circ \)**:
- Similarly, apply the identity:
\[ \cos^2 48^\circ - \sin^2 12^\circ = \cos(48^\circ + 12^\circ) \cos(48^\circ - 12^\circ) = \cos 60^\circ \cos 36^\circ \]

4. **Calculate \( \cos 36^\circ \)**:
- \( \cos 36^\circ \) is known to be \( \frac{\sqrt{5} + 1}{4} \).

5. **Combine the Results**:
- Now we have:
\[ ( \cos 72^\circ \cos 60^\circ )( \cos 60^\circ \cos 36^\circ ) \]
- Substitute the known values:
\[ = \left( \sin 18^\circ \cdot \frac{1}{2} \right) \left( \frac{1}{2} \cdot \frac{\sqrt{5} + 1}{4} \right) \]
- Simplifying gives:
\[ = \frac{1}{4} \cdot \sin 18^\circ \cdot \frac{\sqrt{5} + 1}{4} \]

6. **Final Calculation**:
- We know \( \sin 18^\circ = \frac{\sqrt{5} - 1}{4} \).
- Therefore:
\[ = \frac{1}{4} \cdot \frac{\sqrt{5} - 1}{4} \cdot \frac{\sqrt{5} + 1}{4} \]
- This simplifies to:
\[ = \frac{1}{64} \cdot (5 - 1) = \frac{4}{64} = \frac{1}{16} \]

### Final Answer:
The value of \( ( \cos^2 66^\circ - \sin^2 6^\circ )( \cos^2 48^\circ - \sin^2 12^\circ ) \) is \( \frac{1}{16} \).