Find the absolute value of `(tanAtan2A)+(tan2A tan4A)+(tan4A tanA)` where `A=(2pi)/7`.

Explanation

To solve the problem, we need to find the absolute value of the expression:

\[ \tan A \tan 2A + \tan 2A \tan 4A + \tan 4A \tan A \]

where \( A = \frac{2\pi}{7} \).

### Step 1: Substitute the values of \( A \), \( 2A \), and \( 4A \)

First, we calculate the angles:

- \( A = \frac{2\pi}{7} \)
- \( 2A = \frac{4\pi}{7} \)
- \( 4A = \frac{8\pi}{7} \)

### Step 2: Rewrite the expression using the tangent function

The expression can be rewritten as:

\[ \tan\left(\frac{2\pi}{7}\right) \tan\left(\frac{4\pi}{7}\right) + \tan\left(\frac{4\pi}{7}\right) \tan\left(\frac{8\pi}{7}\right) + \tan\left(\frac{8\pi}{7}\right) \tan\left(\frac{2\pi}{7}\right) \]

### Step 3: Use the property of tangent

We know that:

\[ \tan\left(\frac{8\pi}{7}\right) = \tan\left(\pi + \frac{\pi}{7}\right) = \tan\left(\frac{\pi}{7}\right) \]

Thus, we can rewrite the expression as:

\[ \tan\left(\frac{2\pi}{7}\right) \tan\left(\frac{4\pi}{7}\right) + \tan\left(\frac{4\pi}{7}\right) \tan\left(\frac{\pi}{7}\right) + \tan\left(\frac{\pi}{7}\right) \tan\left(\frac{2\pi}{7}\right) \]

### Step 4: Use the tangent addition formula

We can use the identity for the sum of tangents:

\[ \tan A \tan B + \tan B \tan C + \tan C \tan A = \tan A \tan B \tan C \]

where \( A = \frac{2\pi}{7} \), \( B = \frac{4\pi}{7} \), and \( C = \frac{\pi}{7} \).

### Step 5: Calculate the product of tangents

Using the identity, we find:

\[ \tan\left(\frac{2\pi}{7}\right) \tan\left(\frac{4\pi}{7}\right) \tan\left(\frac{\pi}{7}\right) = \sqrt{7} \]

### Step 6: Substitute back into the expression

Thus, we have:

\[ \tan\left(\frac{2\pi}{7}\right) \tan\left(\frac{4\pi}{7}\right) + \tan\left(\frac{4\pi}{7}\right) \tan\left(\frac{\pi}{7}\right) + \tan\left(\frac{\pi}{7}\right) \tan\left(\frac{2\pi}{7}\right) = \sqrt{7} \]

### Step 7: Find the absolute value

Finally, we need the absolute value of the result:

\[ \text{Absolute value} = |\sqrt{7}| \]

Thus, the final answer is:

\[ \boxed{\sqrt{7}} \]