`tan20^(@)+tan40^(@)+sqrt(3)tan20^(@).tan40^(@)=?`

Options

A. `(sqrt(3))/(4)`
B. `(sqrt(3))/(2)`
C. `sqrt(3)`
D. 1

Answer

To solve the problem \( \tan 20^\circ + \tan 40^\circ + \sqrt{3} \tan 20^\circ \tan 40^\circ \), we can use the tangent addition formula.

### Step-by-Step Solution:

1. **Recognize the Sum of Angles**:
We notice that \( 20^\circ + 40^\circ = 60^\circ \). Therefore, we can rewrite the expression as:
\[ \tan(20^\circ + 40^\circ) = \tan(60^\circ) \]

2. **Use the Tangent Addition Formula**:
The tangent addition formula states:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Here, let \( A = 20^\circ \) and \( B = 40^\circ \). Thus, we have:
\[ \tan(60^\circ) = \frac{\tan 20^\circ + \tan 40^\circ}{1 - \tan 20^\circ \tan 40^\circ} \]

3. **Substituting Known Values**:
We know that \( \tan(60^\circ) = \sqrt{3} \). Therefore, we can set up the equation:
\[ \sqrt{3} = \frac{\tan 20^\circ + \tan 40^\circ}{1 - \tan 20^\circ \tan 40^\circ} \]

4. **Cross-Multiplying**:
Cross-multiplying gives us:
\[ \sqrt{3} (1 - \tan 20^\circ \tan 40^\circ) = \tan 20^\circ + \tan 40^\circ \]

5. **Rearranging the Equation**:
Rearranging the equation, we have:
\[ \tan 20^\circ + \tan 40^\circ + \sqrt{3} \tan 20^\circ \tan 40^\circ = \sqrt{3} \]

6. **Conclusion**:
Therefore, the value of \( \tan 20^\circ + \tan 40^\circ + \sqrt{3} \tan 20^\circ \tan 40^\circ \) is:
\[ \sqrt{3} \]

### Final Answer:
\[ \sqrt{3} \]