If `T` is the period of the function `f(x)=[8x+7]+|tan2pix+cot2pix|-8x]` (where [.] denotes the greatest integer function), then the value of `1/T` is ___________

Explanation

To solve the problem, we need to find the period \( T \) of the function

\[ f(x) = [8x + 7] + |\tan(2\pi x) + \cot(2\pi x)| - 8x \]

where \([.]\) denotes the greatest integer function.

### Step 1: Analyze the periodic components

The function consists of three parts:
1. The greatest integer function \([8x + 7]\)
2. The absolute value function \(|\tan(2\pi x) + \cot(2\pi x)|\)
3. The linear function \(-8x\)

The periodicity of the function will be determined by the periodic components, which in this case are \(\tan(2\pi x)\) and \(\cot(2\pi x)\).

### Step 2: Determine the period of \(\tan(2\pi x)\) and \(\cot(2\pi x)\)

Both \(\tan(kx)\) and \(\cot(kx)\) have a period of \(\frac{\pi}{k}\). For \(\tan(2\pi x)\) and \(\cot(2\pi x)\), the period is:

\[ \text{Period} = \frac{\pi}{2\pi} = \frac{1}{2} \]

### Step 3: Analyze the greatest integer function

The greatest integer function \([8x + 7]\) is not periodic in the traditional sense, but it will change its value at specific intervals. The function \([8x + 7]\) will change its value every time \(8x + 7\) is an integer, which occurs at:

\[ 8x + 7 = n \quad (n \in \mathbb{Z}) \implies x = \frac{n - 7}{8} \]

This means that the function \([8x + 7]\) will change its value every \(\frac{1}{8}\) units.

### Step 4: Find the overall period

The overall period \( T \) of the function \( f(x) \) will be the least common multiple of the periods of the periodic components. The period of \(|\tan(2\pi x) + \cot(2\pi x)|\) is \(\frac{1}{2}\) and the intervals at which \([8x + 7]\) changes is \(\frac{1}{8}\).

To find the least common multiple (LCM):

- The LCM of \(\frac{1}{2}\) and \(\frac{1}{8}\) can be calculated by finding a common denominator. The LCM of \(2\) and \(8\) is \(8\). Thus:

\[ \text{LCM}\left(\frac{1}{2}, \frac{1}{8}\right) = \frac{8}{8} = 1 \]

So, the period \( T = 1 \).

### Step 5: Calculate \( \frac{1}{T} \)

Now we can find \( \frac{1}{T} \):

\[ \frac{1}{T} = \frac{1}{1} = 1 \]

### Final Answer

The value of \( \frac{1}{T} \) is:

\[ \boxed{1} \]