`cos 20^(@) cos 40^(@) cos 80^(@)=`

Options

A. `1/2`
B. `1/4`
C. `1/6`
D. `1/8`

Answer

To solve the expression \( \cos 20^\circ \cos 40^\circ \cos 80^\circ \), we can use a trigonometric identity that simplifies the product of cosines.

### Step-by-Step Solution:

1. **Identify the angles**: We have the angles \( 20^\circ \), \( 40^\circ \), and \( 80^\circ \).

2. **Use the identity**: We can use the identity:
\[ \cos A \cos B \cos C = \frac{1}{4} \left( \cos(A + B + C) + \cos(A + B - C) + \cos(A - B + C) + \cos(-A + B + C) \right) \]
However, a more straightforward approach is to recognize that:
\[ \cos A \cos (60^\circ - A) \cos (60^\circ + A) = \frac{1}{4} \cos 3A \]

3. **Rewrite the angles**: Notice that:
- \( 40^\circ = 60^\circ - 20^\circ \)
- \( 80^\circ = 60^\circ + 20^\circ \)

Thus, we can rewrite the expression as:
\[ \cos 20^\circ \cos 40^\circ \cos 80^\circ = \cos 20^\circ \cos (60^\circ - 20^\circ) \cos (60^\circ + 20^\circ) \]

4. **Apply the identity**: Using the identity mentioned:
\[ \cos 20^\circ \cos 40^\circ \cos 80^\circ = \frac{1}{4} \cos (3 \times 20^\circ) \]

5. **Calculate \( \cos 60^\circ \)**: Now, we find:
\[ \cos 60^\circ = \frac{1}{2} \]

6. **Final calculation**: Substitute back into the equation:
\[ \cos 20^\circ \cos 40^\circ \cos 80^\circ = \frac{1}{4} \cdot \frac{1}{2} = \frac{1}{8} \]

### Conclusion:
Thus, the value of \( \cos 20^\circ \cos 40^\circ \cos 80^\circ \) is \( \frac{1}{8} \).