3.2. Without using a calculator, simplify: \( \cos 20^{\circ} \cos 320^{\circ}-\sin 160^{\circ} \sin 40^{\circ} \)

To simplify \( \cos 20^{\circ} \cos 320^{\circ}-\sin 160^{\circ} \sin 40^{\circ} \), we can use the cosine angle subtraction identity. First, note that \( \cos 320^{\circ} = \cos(360^{\circ} - 40^{\circ}) = \cos 40^{\circ} \). Therefore, the expression becomes: \[ \cos 20^{\circ} \cos 40^{\circ} - \sin 160^{\circ} \sin 40^{\circ} \] Next, we recognize that \( \sin 160^{\circ} = \sin(180^{\circ} - 20^{\circ}) = \sin 20^{\circ} \). Thus, we can rewrite the expression as: \[ \cos 20^{\circ} \cos 40^{\circ} - \sin 20^{\circ} \sin 40^{\circ} \] Now, we can apply the cosine subtraction formula: \[ \cos A \cos B - \sin A \sin B = \cos(A + B) \] By setting \( A = 20^{\circ} \) and \( B = 40^{\circ} \), we get: \[ \cos(20^{\circ} + 40^{\circ}) = \cos 60^{\circ} \] Since \( \cos 60^{\circ} = \frac{1}{2} \), the original expression simplifies to: \[ \frac{1}{2} \] Therefore, the final simplified result is: \[ \boxed{\frac{1}{2}} \]