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

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:

\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]

We will apply this formula to the second and third terms in the equation.

### Step 2: Expand \(\tan\left(\theta + \frac{\pi}{3}\right)\) and \(\tan\left(\theta + \frac{2\pi}{3}\right)\)
Using the tangent addition formula:

1. For \(\tan\left(\theta + \frac{\pi}{3}\right)\):
\[ \tan\left(\theta + \frac{\pi}{3}\right) = \frac{\tan \theta + \tan\frac{\pi}{3}}{1 - \tan \theta \tan\frac{\pi}{3}} = \frac{\tan \theta + \sqrt{3}}{1 - \tan \theta \sqrt{3}} \]

2. For \(\tan\left(\theta + \frac{2\pi}{3}\right)\):
\[ \tan\left(\theta + \frac{2\pi}{3}\right) = \frac{\tan \theta + \tan\frac{2\pi}{3}}{1 - \tan \theta \tan\frac{2\pi}{3}} = \frac{\tan \theta - \sqrt{3}}{1 + \tan \theta \sqrt{3}} \]

### Step 3: Substitute back into the equation
Now, substituting these expansions into the original equation gives:

\[ \tan \theta + \frac{\tan \theta + \sqrt{3}}{1 - \tan \theta \sqrt{3}} + \frac{\tan \theta - \sqrt{3}}{1 + \tan \theta \sqrt{3}} = 3 \]

### Step 4: Combine the fractions
Let \(x = \tan \theta\). The equation becomes:

\[ x + \frac{x + \sqrt{3}}{1 - x\sqrt{3}} + \frac{x - \sqrt{3}}{1 + x\sqrt{3}} = 3 \]

To combine the fractions, we will find a common denominator:

\[ \text{Common Denominator} = (1 - x\sqrt{3})(1 + x\sqrt{3}) = 1 - 3x^2 \]

### Step 5: Rewrite the equation
Now, rewriting the equation with the common denominator:

\[ x(1 - 3x^2) + (x + \sqrt{3})(1 + x\sqrt{3}) + (x - \sqrt{3})(1 - x\sqrt{3}) = 3(1 - 3x^2) \]

### Step 6: Simplify the equation
After simplifying the left-hand side and setting it equal to the right-hand side, we will collect like terms and rearrange:

\[ 3x - 3x^3 + 2\sqrt{3}x = 3 - 9x^2 \]

### Step 7: Rearranging and solving for \(x\)
This leads to a polynomial equation in \(x\):

\[ 3x^3 - 9x^2 + (3 - 2\sqrt{3})x + 3 = 0 \]

### Step 8: Use the cubic formula or numerical methods
To find the roots of this cubic equation, we can use numerical methods or the cubic formula.

### Step 9: Find \(\theta\)
Once we have the values of \(x = \tan \theta\), we can find \(\theta\) using:

\[ \theta = \tan^{-1}(x) + n\pi \quad (n \in \mathbb{Z}) \]

### Final Answer
The final solution will be:

\[ \theta = n\pi + \frac{\pi}{12} \quad (n \in \mathbb{Z}) \]