If `A, B and C` are three values lying in `[0, 2pi]` for which `tan theta = K` then `tan""(A)/(3) tan ""(B)/(3) + tan ""(B)/(3)tan ""(C )/(3) + tan ""(C )/(3)tan ""(A)/(3)` is equal to ___________.

Explanation

To solve the problem, we need to evaluate the expression:

\[ \frac{\tan A}{3} \cdot \frac{\tan B}{3} + \frac{\tan B}{3} \cdot \frac{\tan C}{3} + \frac{\tan C}{3} \cdot \frac{\tan A}{3} \]

Let's denote \( x = \frac{\tan A}{3} \), \( y = \frac{\tan B}{3} \), and \( z = \frac{\tan C}{3} \). Thus, we can rewrite the expression as:

\[ xy + yz + zx \]

### Step 1: Relate \( \tan A, \tan B, \tan C \) to \( K \)

Given that \( \tan \theta = K \), we can express \( \tan A, \tan B, \tan C \) in terms of \( K \). We know that:

\[ \tan A = 3x, \quad \tan B = 3y, \quad \tan C = 3z \]

### Step 2: Use the identity for \( \tan 3\theta \)

Using the identity for \( \tan 3\theta \):

\[ \tan 3\theta = \frac{3\tan \theta - \tan^3 \theta}{1 - 3\tan^2 \theta} \]

Substituting \( \tan \theta = K \):

\[ \tan 3\theta = \frac{3K - K^3}{1 - 3K^2} \]

### Step 3: Set up the cubic equation

Assuming \( \tan A, \tan B, \tan C \) are the roots of the cubic equation derived from the identity, we can express it as:

\[ t^3 - 3K t^2 + 3K t - K = 0 \]

### Step 4: Identify the coefficients

From the cubic equation \( t^3 + pt^2 + qt + r = 0 \), we can identify:

- The sum of the roots \( (A + B + C) = 3K \)
- The sum of the product of the roots taken two at a time \( (AB + BC + CA) = 3K \)
- The product of the roots \( (ABC) = K \)

### Step 5: Calculate \( xy + yz + zx \)

From the above, we know:

\[ xy + yz + zx = \frac{1}{9}(AB + BC + CA) = \frac{1}{9}(3K) = \frac{K}{3} \]

### Final Result

Thus, the expression simplifies to:

\[ \frac{K}{3} \]

### Summary of Steps

1. Define \( x, y, z \) in terms of \( \tan A, \tan B, \tan C \).
2. Use the identity for \( \tan 3\theta \) to relate it to \( K \).
3. Set up the cubic equation with roots \( \tan A, \tan B, \tan C \).
4. Identify the coefficients of the cubic equation.
5. Calculate \( xy + yz + zx \) using the identified coefficients.