If `2 sin""A/2=sqrt(1+sinA)+sqrt(1-sinA,)then A/2` lies between,

Options

A. `2npi+(pi)/(4)and 2npi+(3pi)/(4),n inZ`
B. `2npi-(pi)/(4)and 2npi+(pi)/(4),n inZ`
C. `2npi-(3pi)/(4)and 2npi-(pi)/(4),n inZ`
D. `-ooand +oo`

Answer

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

### Step 1: Simplify the Right-Hand Side
We know that:
\[ \sqrt{1 + \sin A} + \sqrt{1 - \sin A} \]
can be simplified using the identity for sine. We can express \(1 + \sin A\) and \(1 - \sin A\) in terms of cosine:
\[ \sqrt{1 + \sin A} = \sqrt{(\sin A + 1)} \quad \text{and} \quad \sqrt{1 - \sin A} = \sqrt{(\cos^2 A)} = \cos A \]
Thus, we can rewrite:
\[ \sqrt{1 + \sin A} + \sqrt{1 - \sin A} = \sqrt{(1 + \sin A)} + \sqrt{(1 - \sin A)} \]

### Step 2: Use Trigonometric Identities
Using the identity \(1 - \sin^2 A = \cos^2 A\), we can express the right-hand side as:
\[ \sqrt{1 + \sin A} + \sqrt{1 - \sin A} = \sqrt{(1 + \sin A)} + \sqrt{(1 - \sin A)} = \sqrt{(1 + \sin A)(1 - \sin A)} = \sqrt{\cos^2 A} \]

### Step 3: Set Up the Equation
Now, we have:
\[ 2 \sin \frac{A}{2} = \sqrt{1 + \sin A} + \sqrt{1 - \sin A} \]
This simplifies to:
\[ 2 \sin \frac{A}{2} = \sqrt{2(1 + \sin A)} \]

### Step 4: Square Both Sides
To eliminate the square root, we square both sides:
\[ (2 \sin \frac{A}{2})^2 = (1 + \sin A) \]
This gives:
\[ 4 \sin^2 \frac{A}{2} = 1 + \sin A \]

### Step 5: Use the Double Angle Identity
Recall that \(\sin A = 2 \sin \frac{A}{2} \cos \frac{A}{2}\):
\[ 4 \sin^2 \frac{A}{2} = 1 + 2 \sin \frac{A}{2} \cos \frac{A}{2} \]

### Step 6: Rearranging the Equation
Rearranging gives us:
\[ 4 \sin^2 \frac{A}{2} - 2 \sin \frac{A}{2} \cos \frac{A}{2} - 1 = 0 \]
Let \(x = \sin \frac{A}{2}\), then we have:
\[ 4x^2 - 2x \sqrt{1 - x^2} - 1 = 0 \]

### Step 7: Solve the Quadratic Equation
This is a quadratic in \(x\) that can be solved using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 4\), \(b = -2\sqrt{1 - x^2}\), and \(c = -1\).

### Step 8: Determine the Range of \(\frac{A}{2}\)
From the solutions obtained, we can find the range of \(\frac{A}{2}\). The values of \(\frac{A}{2}\) will lie between \(0\) and \(\frac{\pi}{2}\) based on the conditions derived from the trigonometric identities.

### Final Answer
Thus, \(\frac{A}{2}\) lies between:
\[ \left[\frac{\pi}{4}, \frac{\pi}{2}\right] \]