Solve `tan x=[x], x in (0, 3pi//2)`. Here [.] represents the greatest integer function.

Explanation

To solve the equation \( \tan x = [x] \) where \( x \) is in the interval \( (0, \frac{3\pi}{2}) \) and \([x]\) represents the greatest integer function, we can follow these steps:

### Step 1: Determine the range of \([x]\)
In the interval \( (0, \frac{3\pi}{2}) \):
- The value of \( \frac{3\pi}{2} \) is approximately \( 4.71 \).
- Therefore, \([x]\) can take values \( 0, 1, 2, 3, \) and \( 4 \).

### Step 2: Analyze each case for \([x]\)
We will analyze the equation \( \tan x = n \) for \( n = 0, 1, 2, 3, 4 \).

#### Case 1: \( [x] = 0 \)
- Here, \( \tan x = 0 \).
- The solutions are \( x = 0, \pi, 2\pi, \ldots \)
- In the interval \( (0, \frac{3\pi}{2}) \), the only solution is \( x = \pi \).

#### Case 2: \( [x] = 1 \)
- Here, \( \tan x = 1 \).
- The solutions are \( x = \frac{\pi}{4} + n\pi \).
- In the interval \( (0, \frac{3\pi}{2}) \), the solution is \( x = \frac{\pi}{4} \) and \( x = \frac{5\pi}{4} \).

#### Case 3: \( [x] = 2 \)
- Here, \( \tan x = 2 \).
- The solutions are \( x = \tan^{-1}(2) + n\pi \).
- In the interval \( (0, \frac{3\pi}{2}) \), the solution is \( x = \tan^{-1}(2) \) and \( x = \tan^{-1}(2) + \pi \).

#### Case 4: \( [x] = 3 \)
- Here, \( \tan x = 3 \).
- The solutions are \( x = \tan^{-1}(3) + n\pi \).
- In the interval \( (0, \frac{3\pi}{2}) \), the solution is \( x = \tan^{-1}(3) \) and \( x = \tan^{-1}(3) + \pi \).

#### Case 5: \( [x] = 4 \)
- Here, \( \tan x = 4 \).
- The solutions are \( x = \tan^{-1}(4) + n\pi \).
- In the interval \( (0, \frac{3\pi}{2}) \), the solution is \( x = \tan^{-1}(4) \).

### Step 3: Compile all solutions
Now we compile all the solutions found:
1. From \( [x] = 0 \): \( x = \pi \)
2. From \( [x] = 1 \): \( x = \frac{\pi}{4}, \frac{5\pi}{4} \)
3. From \( [x] = 2 \): \( x = \tan^{-1}(2), \tan^{-1}(2) + \pi \)
4. From \( [x] = 3 \): \( x = \tan^{-1}(3), \tan^{-1}(3) + \pi \)
5. From \( [x] = 4 \): \( x = \tan^{-1}(4) \)

### Final Answer
The solutions to the equation \( \tan x = [x] \) in the interval \( (0, \frac{3\pi}{2}) \) are:
- \( x = \pi \)
- \( x = \frac{\pi}{4} \)
- \( x = \frac{5\pi}{4} \)
- \( x = \tan^{-1}(2) \)
- \( x = \tan^{-1}(3) \)
- \( x = \tan^{-1}(4) \)