If ` (cos^(3) theta )/( (1- sin theta ))+( sin^(3) theta )/( (1+ cos theta ))=1+ cos theta `, then number of possible values of ` theta ` is ( where ` theta in [0,2pi]`)

Answer

To solve the equation

\[ \frac{\cos^3 \theta}{1 - \sin \theta} + \frac{\sin^3 \theta}{1 + \cos \theta} = 1 + \cos \theta \]

for \( \theta \) in the interval \([0, 2\pi]\), we will follow these steps:

### Step 1: Rationalize the fractions
We start by rationalizing the two fractions separately.

1. The first term:
\[ \frac{\cos^3 \theta}{1 - \sin \theta} \cdot \frac{1 + \sin \theta}{1 + \sin \theta} = \frac{\cos^3 \theta (1 + \sin \theta)}{(1 - \sin \theta)(1 + \sin \theta)} = \frac{\cos^3 \theta (1 + \sin \theta)}{\cos^2 \theta} \]
This simplifies to:
\[ \frac{\cos \theta (1 + \sin \theta)}{1} \]

2. The second term:
\[ \frac{\sin^3 \theta}{1 + \cos \theta} \cdot \frac{1 - \cos \theta}{1 - \cos \theta} = \frac{\sin^3 \theta (1 - \cos \theta)}{(1 + \cos \theta)(1 - \cos \theta)} = \frac{\sin^3 \theta (1 - \cos \theta)}{\sin^2 \theta} \]
This simplifies to:
\[ \frac{\sin \theta (1 - \cos \theta)}{1} \]

### Step 2: Combine the terms
Now we can combine the two simplified terms:
\[ \cos \theta (1 + \sin \theta) + \sin \theta (1 - \cos \theta) = 1 + \cos \theta \]

### Step 3: Expand and simplify
Expanding the left side:
\[ \cos \theta + \cos \theta \sin \theta + \sin \theta - \sin \theta \cos \theta = 1 + \cos \theta \]
This simplifies to:
\[ \cos \theta + \sin \theta = 1 + \cos \theta \]

### Step 4: Isolate the sine term
Subtract \(\cos \theta\) from both sides:
\[ \sin \theta = 1 \]

### Step 5: Solve for \(\theta\)
The equation \(\sin \theta = 1\) has a solution:
\[ \theta = \frac{\pi}{2} \]

### Step 6: Check for validity
We need to check if this solution is valid in the original equation:
Substituting \(\theta = \frac{\pi}{2}\):
- \(\cos \frac{\pi}{2} = 0\)
- \(\sin \frac{\pi}{2} = 1\)

Substituting into the left side:
\[ \frac{0^3}{1 - 1} + \frac{1^3}{1 + 0} = \frac{0}{0} + 1 = \text{undefined} \]
The left side is indeterminate, thus \(\theta = \frac{\pi}{2}\) is rejected.

### Conclusion
Since there are no other solutions in the interval \([0, 2\pi]\), the number of possible values of \(\theta\) is:

\[ \boxed{0} \]