`tan 20^(@) + tan40^(@) + sqrt(3).tan 20^(@).tan 40^(@)=`

Options

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

Answer

To solve the expression \( \tan 20^\circ + \tan 40^\circ + \sqrt{3} \tan 20^\circ \tan 40^\circ \), we can use the tangent addition formula. Here’s a step-by-step breakdown:

### Step 1: Recognize the angle addition
We can rewrite the expression using the identity for \( \tan(A + B) \):
\[ \tan(60^\circ) = \tan(20^\circ + 40^\circ) \]

### Step 2: Apply the tangent addition formula
The formula for \( \tan(A + B) \) is:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Setting \( A = 20^\circ \) and \( B = 40^\circ \), we have:
\[ \tan(60^\circ) = \frac{\tan 20^\circ + \tan 40^\circ}{1 - \tan 20^\circ \tan 40^\circ} \]

### Step 3: Substitute known values
We know that \( \tan(60^\circ) = \sqrt{3} \). Therefore, we can write:
\[ \sqrt{3} = \frac{\tan 20^\circ + \tan 40^\circ}{1 - \tan 20^\circ \tan 40^\circ} \]

### Step 4: Rearranging the equation
Cross-multiplying gives:
\[ \sqrt{3}(1 - \tan 20^\circ \tan 40^\circ) = \tan 20^\circ + \tan 40^\circ \]
This simplifies to:
\[ \sqrt{3} - \sqrt{3} \tan 20^\circ \tan 40^\circ = \tan 20^\circ + \tan 40^\circ \]

### Step 5: Rearranging to find the expression
Now, we can rearrange this to isolate \( \tan 20^\circ + \tan 40^\circ + \sqrt{3} \tan 20^\circ \tan 40^\circ \):
\[ \tan 20^\circ + \tan 40^\circ + \sqrt{3} \tan 20^\circ \tan 40^\circ = \sqrt{3} \]

### Conclusion
Thus, the value of the expression \( \tan 20^\circ + \tan 40^\circ + \sqrt{3} \tan 20^\circ \tan 40^\circ \) is:
\[ \sqrt{3} \]