The value of ` tan 40^(@) + tan 20^(@) + sqrt(3) tan 20^(@) tan 40^(@)` is

Options

A. `sqrt(12)`
B. `(1)/(sqrt(3))`
C. 1
D. `sqrt(3)`

Answer

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

### Step-by-step Solution:

1. **Recall the tangent addition formula**:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Here, we can set \( A = 20^\circ \) and \( B = 40^\circ \).

2. **Apply the formula for \( \tan(60^\circ) \)**:
Since \( 60^\circ = 20^\circ + 40^\circ \), we can write:
\[ \tan 60^\circ = \frac{\tan 20^\circ + \tan 40^\circ}{1 - \tan 20^\circ \tan 40^\circ} \]

3. **Substitute the known value of \( \tan 60^\circ \)**:
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-multiply to eliminate the fraction**:
\[ \sqrt{3}(1 - \tan 20^\circ \tan 40^\circ) = \tan 20^\circ + \tan 40^\circ \]

5. **Rearranging the equation**:
Expanding and rearranging gives:
\[ \sqrt{3} - \sqrt{3} \tan 20^\circ \tan 40^\circ = \tan 20^\circ + \tan 40^\circ \]

6. **Combine like terms**:
Now, we can rearrange this to isolate \( \tan 20^\circ + \tan 40^\circ \):
\[ \tan 20^\circ + \tan 40^\circ + \sqrt{3} \tan 20^\circ \tan 40^\circ = \sqrt{3} \]

7. **Conclusion**:
Thus, we have:
\[ \tan 40^\circ + \tan 20^\circ + \sqrt{3} \tan 20^\circ \tan 40^\circ = \sqrt{3} \]

### Final Answer:
The value is \( \sqrt{3} \).

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