If `0 leq theta leq pi and sin(theta/2)=sqrt(1+sintheta)-sqrt(1-sintheta)` , then the possible value of `tan theta` is (a) `4/3` (b) 0 (c) `-3/4` (d) `-4/3`

Options

A. `(4)/(3)`
B. 0
C. `(-3)/(4)`
D. `(-4)/(3)`

Explanation

To solve the equation \( \sin\left(\frac{\theta}{2}\right) = \sqrt{1 + \sin \theta} - \sqrt{1 - \sin \theta} \), we will follow these steps:

### Step 1: Square both sides
Start by squaring both sides of the equation to eliminate the square roots.

\[ \sin^2\left(\frac{\theta}{2}\right) = \left(\sqrt{1 + \sin \theta} - \sqrt{1 - \sin \theta}\right)^2 \]

### Step 2: Expand the right side
Using the identity \( (a - b)^2 = a^2 - 2ab + b^2 \):

\[ \sin^2\left(\frac{\theta}{2}\right) = (1 + \sin \theta) + (1 - \sin \theta) - 2\sqrt{(1 + \sin \theta)(1 - \sin \theta)} \]

This simplifies to:

\[ \sin^2\left(\frac{\theta}{2}\right) = 2 - 2\sqrt{1 - \sin^2 \theta} \]

### Step 3: Use the Pythagorean identity
Recall that \( \sin^2 \theta + \cos^2 \theta = 1 \), so \( \sqrt{1 - \sin^2 \theta} = \cos \theta \):

\[ \sin^2\left(\frac{\theta}{2}\right) = 2 - 2\cos \theta \]

### Step 4: Express \( \sin^2\left(\frac{\theta}{2}\right) \)
Using the half-angle identity, we know that:

\[ \sin^2\left(\frac{\theta}{2}\right) = \frac{1 - \cos \theta}{2} \]

### Step 5: Set the equations equal
Now we can set the two expressions for \( \sin^2\left(\frac{\theta}{2}\right) \) equal to each other:

\[ \frac{1 - \cos \theta}{2} = 2 - 2\cos \theta \]

### Step 6: Clear the fraction
Multiply through by 2 to eliminate the fraction:

\[ 1 - \cos \theta = 4 - 4\cos \theta \]

### Step 7: Rearrange the equation
Rearranging gives:

\[ 4\cos \theta - \cos \theta = 4 - 1 \]

\[ 3\cos \theta = 3 \]

### Step 8: Solve for \( \cos \theta \)
Dividing both sides by 3:

\[ \cos \theta = 1 \]

### Step 9: Find \( \theta \)
Since \( \cos \theta = 1 \) implies \( \theta = 0 \) (within the range \( 0 \leq \theta \leq \pi \)).

### Step 10: Calculate \( \tan \theta \)
Now, we find \( \tan \theta \):

\[ \tan \theta = \tan(0) = 0 \]

### Conclusion
Thus, the possible value of \( \tan \theta \) is:

\[ \boxed{0} \]