Find number of solutions of equation
`sin^(2) theta- 4/(sin^(3) theta-1)=1- 4/(sin^(3) theta-1), theta in [0, 6pi]`.

Explanation

To solve the equation

\[ \sin^2 \theta - \frac{4}{\sin^3 \theta - 1} = 1 - \frac{4}{\sin^3 \theta - 1} \]

for \(\theta\) in the interval \([0, 6\pi]\), we can follow these steps:

### Step 1: Simplify the equation
First, we can simplify the given equation. By rearranging the terms, we have:

\[ \sin^2 \theta - \frac{4}{\sin^3 \theta - 1} = 1 - \frac{4}{\sin^3 \theta - 1} \]

This implies:

\[ \sin^2 \theta - 1 = 0 \]

### Step 2: Solve for \(\sin^2 \theta\)
From the equation \(\sin^2 \theta - 1 = 0\), we can deduce:

\[ \sin^2 \theta = 1 \]

### Step 3: Find values of \(\theta\)
The solutions for \(\sin^2 \theta = 1\) occur when:

\[ \sin \theta = 1 \quad \text{or} \quad \sin \theta = -1 \]

The angle \(\theta\) for which \(\sin \theta = 1\) is:

\[ \theta = \frac{\pi}{2} + 2k\pi \quad \text{for integers } k \]

The angle \(\theta\) for which \(\sin \theta = -1\) is:

\[ \theta = \frac{3\pi}{2} + 2k\pi \quad \text{for integers } k \]

### Step 4: Determine the values of \(k\) for the interval \([0, 6\pi]\)
Now we need to find all possible values of \(\theta\) in the interval \([0, 6\pi]\).

1. For \(\sin \theta = 1\):
- When \(k = 0\): \(\theta = \frac{\pi}{2}\)
- When \(k = 1\): \(\theta = \frac{\pi}{2} + 2\pi = \frac{5\pi}{2}\)
- When \(k = 2\): \(\theta = \frac{\pi}{2} + 4\pi = \frac{9\pi}{2}\) (not in the interval)

2. For \(\sin \theta = -1\):
- When \(k = 0\): \(\theta = \frac{3\pi}{2}\)
- When \(k = 1\): \(\theta = \frac{3\pi}{2} + 2\pi = \frac{7\pi}{2}\)
- When \(k = 2\): \(\theta = \frac{3\pi}{2} + 4\pi = \frac{11\pi}{2}\) (not in the interval)

### Step 5: List all the solutions
Thus, the solutions in the interval \([0, 6\pi]\) are:

- From \(\sin \theta = 1\): \(\frac{\pi}{2}, \frac{5\pi}{2}\)
- From \(\sin \theta = -1\): \(\frac{3\pi}{2}, \frac{7\pi}{2}\)

### Step 6: Count the solutions
The valid solutions in the interval \([0, 6\pi]\) are:

1. \(\frac{\pi}{2}\)
2. \(\frac{3\pi}{2}\)
3. \(\frac{5\pi}{2}\)
4. \(\frac{7\pi}{2}\)

Thus, we have a total of **4 solutions**.

### Final Answer
The number of solutions of the equation in the interval \([0, 6\pi]\) is **4**.

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