The value of `(1+tan\ (3pi)/8*tan\ pi/8)+(1+tan\ (5pi)/8*tan\ (3pi)/8)+(1+tan\ (7pi)/8*tan\ (5pi)/8)+(1+tan\ (9pi)/8* tan\ (7pi)/8)`

Answer

To solve the expression

\[ (1 + \tan(3\pi/8) \tan(\pi/8)) + (1 + \tan(5\pi/8) \tan(3\pi/8)) + (1 + \tan(7\pi/8) \tan(5\pi/8)) + (1 + \tan(9\pi/8) \tan(7\pi/8)), \]

we can simplify each term using the identity for the tangent of the difference of two angles.

### Step 1: Rewrite each term
We can use the identity:

\[ 1 + \tan A \tan B = \frac{\tan(A - B)}{\tan(A - B)} + 1 = \frac{\tan(A - B) + 1}{\tan(A - B)}. \]

Thus, we rewrite each term as follows:

1. \(1 + \tan(3\pi/8) \tan(\pi/8)\)
2. \(1 + \tan(5\pi/8) \tan(3\pi/8)\)
3. \(1 + \tan(7\pi/8) \tan(5\pi/8)\)
4. \(1 + \tan(9\pi/8) \tan(7\pi/8)\)

### Step 2: Use the tangent subtraction identity
Using the identity \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \), we can simplify each term.

For example, for the first term:

\[ 1 + \tan(3\pi/8) \tan(\pi/8) = \frac{\tan(3\pi/8) - \tan(\pi/8)}{\tan(3\pi/8) - \tan(\pi/8)} + 1 = \tan(3\pi/8) - \tan(\pi/8) + 1. \]

### Step 3: Combine the terms
Now we can combine all the terms:

\[ (\tan(3\pi/8) - \tan(\pi/8) + 1) + (\tan(5\pi/8) - \tan(3\pi/8) + 1) + (\tan(7\pi/8) - \tan(5\pi/8) + 1) + (\tan(9\pi/8) - \tan(7\pi/8) + 1). \]

Notice that the \(\tan\) terms will cancel out:

\[ \tan(3\pi/8) - \tan(\pi/8) + \tan(5\pi/8) - \tan(3\pi/8) + \tan(7\pi/8) - \tan(5\pi/8) + \tan(9\pi/8) - \tan(7\pi/8). \]

### Step 4: Simplify further
The terms cancel out, leading to:

\[ \tan(9\pi/8) - \tan(\pi/8). \]

### Step 5: Evaluate \(\tan(9\pi/8)\)
Since \(9\pi/8\) is in the third quadrant, we have:

\[ \tan(9\pi/8) = \tan(\pi + \pi/8) = \tan(\pi/8). \]

Thus,

\[ \tan(9\pi/8) - \tan(\pi/8) = \tan(\pi/8) - \tan(\pi/8) = 0. \]

### Final Answer
The value of the entire expression is

\[ \boxed{0}. \]