If `alpha,beta,gamma,delta` are the four solutions of the equation `tan(theta+pi/4)=3 tan 3theta.` No two of which have equal tangents, then the value of `tan alpha+tan beta+tan gamma+tan delta=`

Options

A. `1//3`
B. `8//3`
C. `-8//3`
D. 0

Explanation

To solve the equation \( \tan(\theta + \frac{\pi}{4}) = 3 \tan(3\theta) \) and find the value of \( \tan \alpha + \tan \beta + \tan \gamma + \tan \delta \), we will follow these steps:

### Step 1: Rewrite the left-hand side using the tangent addition formula
Using the tangent addition formula, we can rewrite the left-hand side:
\[ \tan(\theta + \frac{\pi}{4}) = \frac{\tan \theta + 1}{1 - \tan \theta} \]

### Step 2: Rewrite the right-hand side
The right-hand side can be expressed as:
\[ 3 \tan(3\theta) = 3 \cdot \frac{3\tan \theta - \tan^3 \theta}{1 - 3\tan^2 \theta} \]

### Step 3: Set the two sides equal
Now, we set the two expressions equal to each other:
\[ \frac{\tan \theta + 1}{1 - \tan \theta} = 3 \cdot \frac{3\tan \theta - \tan^3 \theta}{1 - 3\tan^2 \theta} \]

### Step 4: Cross-multiply
Cross-multiplying gives:
\[ (\tan \theta + 1)(1 - 3\tan^2 \theta) = 3(3\tan \theta - \tan^3 \theta)(1 - \tan \theta) \]

### Step 5: Expand both sides
Expanding both sides results in:
\[ \tan \theta + 1 - 3\tan^3 \theta - 3\tan^2 \theta = 9\tan \theta - 3\tan^3 \theta - 9\tan^2 \theta + 3\tan^4 \theta \]

### Step 6: Rearranging the equation
Rearranging gives us:
\[ 3\tan^4 \theta - 6\tan^2 \theta + 8\tan \theta - 1 = 0 \]

### Step 7: Identify the coefficients
This is a polynomial equation in terms of \( \tan \theta \) of degree 4. The roots of this polynomial are \( \tan \alpha, \tan \beta, \tan \gamma, \tan \delta \).

### Step 8: Use Vieta's formulas
According to Vieta's formulas, the sum of the roots of the polynomial \( ax^4 + bx^3 + cx^2 + dx + e = 0 \) is given by:
\[ -\frac{b}{a} \]
In our case, \( a = 3 \) and \( b = 0 \) (since there is no \( \tan^3 \theta \) term). Therefore:
\[ \tan \alpha + \tan \beta + \tan \gamma + \tan \delta = -\frac{0}{3} = 0 \]

### Final Answer
Thus, the value of \( \tan \alpha + \tan \beta + \tan \gamma + \tan \delta \) is:
\[ \boxed{0} \]