The value of `tan 20^(@) tan 40^(@) tan 80^(@)` is equal to

Options

A. `tan 60^(@)`
B. `cot 60^(@)`
C. `tan 45^(@)`
D. `tan 80^(@)`

Explanation

To find the value of \( \tan 20^\circ \tan 40^\circ \tan 80^\circ \), we can use the properties of tangent and some trigonometric identities.

### Step-by-Step Solution:

1. **Express \( \tan 40^\circ \) and \( \tan 80^\circ \)**:
We can rewrite \( \tan 40^\circ \) and \( \tan 80^\circ \) using the tangent subtraction and addition formulas:
\[ \tan 40^\circ = \tan(60^\circ - 20^\circ) \]
\[ \tan 80^\circ = \tan(60^\circ + 20^\circ) \]

2. **Use the tangent subtraction and addition formulas**:
Using the tangent subtraction and addition formulas:
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
We can set \( A = 60^\circ \) and \( B = 20^\circ \):
\[ \tan 40^\circ = \frac{\tan 60^\circ - \tan 20^\circ}{1 + \tan 60^\circ \tan 20^\circ} \]
\[ \tan 80^\circ = \frac{\tan 60^\circ + \tan 20^\circ}{1 - \tan 60^\circ \tan 20^\circ} \]

3. **Substituting \( \tan 60^\circ \)**:
We know that \( \tan 60^\circ = \sqrt{3} \). So we can substitute this value into our equations:
\[ \tan 40^\circ = \frac{\sqrt{3} - \tan 20^\circ}{1 + \sqrt{3} \tan 20^\circ} \]
\[ \tan 80^\circ = \frac{\sqrt{3} + \tan 20^\circ}{1 - \sqrt{3} \tan 20^\circ} \]

4. **Multiply all three tangents**:
Now we can multiply \( \tan 20^\circ \), \( \tan 40^\circ \), and \( \tan 80^\circ \):
\[ \tan 20^\circ \tan 40^\circ \tan 80^\circ = \tan 20^\circ \cdot \left( \frac{\sqrt{3} - \tan 20^\circ}{1 + \sqrt{3} \tan 20^\circ} \right) \cdot \left( \frac{\sqrt{3} + \tan 20^\circ}{1 - \sqrt{3} \tan 20^\circ} \right) \]

5. **Using the property of tangent**:
There is a known property:
\[ \tan A \tan(60^\circ - A) \tan(60^\circ + A) = \tan 3A \]
Applying this property with \( A = 20^\circ \):
\[ \tan 20^\circ \tan 40^\circ \tan 80^\circ = \tan 60^\circ \]

6. **Final answer**:
Since \( \tan 60^\circ = \sqrt{3} \), we conclude:
\[ \tan 20^\circ \tan 40^\circ \tan 80^\circ = \sqrt{3} \]

### Final Result:
The value of \( \tan 20^\circ \tan 40^\circ \tan 80^\circ \) is \( \sqrt{3} \).