Solve that equations :
`tantheta + tan(theta+(pi)/3) + tan(theta+(2pi)/3)=3`

Options

A. `theta = npi+pi/2`
B. `theta = (npi)/6+pi/2`
C. `theta = (npi)/6+pi/12`
D. `theta = (npi)/3+pi/12`

Explanation

To solve the equation \( \tan \theta + \tan \left( \theta + \frac{\pi}{3} \right) + \tan \left( \theta + \frac{2\pi}{3} \right) = 3 \), we will follow these steps:

### Step 1: Use the tangent addition formula
We know that:
\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
Using this, we can express \( \tan \left( \theta + \frac{\pi}{3} \right) \) and \( \tan \left( \theta + \frac{2\pi}{3} \right) \).

### Step 2: Calculate \( \tan \left( \theta + \frac{\pi}{3} \right) \)
Using the addition formula:
\[ \tan \left( \theta + \frac{\pi}{3} \right) = \frac{\tan \theta + \sqrt{3}}{1 - \tan \theta \cdot \sqrt{3}} \]

### Step 3: Calculate \( \tan \left( \theta + \frac{2\pi}{3} \right) \)
Similarly, we have:
\[ \tan \left( \theta + \frac{2\pi}{3} \right) = \frac{\tan \theta - \sqrt{3}}{1 + \tan \theta \cdot \sqrt{3}} \]

### Step 4: Substitute back into the equation
Now we substitute these values back into the original equation:
\[ \tan \theta + \frac{\tan \theta + \sqrt{3}}{1 - \tan \theta \cdot \sqrt{3}} + \frac{\tan \theta - \sqrt{3}}{1 + \tan \theta \cdot \sqrt{3}} = 3 \]

### Step 5: Combine the fractions
To combine the fractions, we need a common denominator:
\[ (1 - \tan \theta \cdot \sqrt{3})(1 + \tan \theta \cdot \sqrt{3}) \]
This simplifies to:
\[ 1 - 3 \tan^2 \theta \]
Thus, we rewrite the equation as:
\[ \tan \theta (1 - 3 \tan^2 \theta) + (\tan \theta + \sqrt{3})(1 + \tan \theta \cdot \sqrt{3}) + (\tan \theta - \sqrt{3})(1 - \tan \theta \cdot \sqrt{3}) = 3(1 - 3 \tan^2 \theta) \]

### Step 6: Simplify the equation
After simplifying, we get:
\[ 9 \tan \theta - 3 \tan^3 \theta = 3(1 - 3 \tan^2 \theta) \]
Rearranging gives:
\[ 3 \tan^3 \theta - 9 \tan \theta + 9 = 0 \]

### Step 7: Factor the cubic equation
Factoring out 3, we have:
\[ \tan^3 \theta - 3 \tan \theta + 3 = 0 \]

### Step 8: Solve for \( \tan \theta \)
Let \( x = \tan \theta \). We can use numerical methods or synthetic division to find the roots of this cubic equation.

### Step 9: Find the general solution
Once we find the values of \( x \), we can find \( \theta \) using:
\[ \theta = \tan^{-1}(x) + n\pi, \quad n \in \mathbb{Z} \]

### Final Solution
The final solution will be in the form:
\[ \theta = n\frac{\pi}{3} + \frac{\pi}{12}, \quad n \in \mathbb{Z} \]