If `2 cos ( A//2) = sqrt(( 1+ sin A )) - sqrt(( 1- sin A )) `, then

Options

A. `npi + ( pi //4) lt A//2 lt npi + ( 3pi //4)`
B. `npi - ( pi //4) lt A//2 lt 2npi - ( 3pi //4)`
C. `2npi - ( 3pi //4) lt A//2 lt 2npi + ( 5pi //4)`
D. `2npi + ( pi //4) lt A//2 lt 2npi + ( 3pi //4)`

Answer

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

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

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

### Step 2: Simplify the left-hand side
The left-hand side simplifies to:

\[ 4 \cos^2 \left( \frac{A}{2} \right) \]

### Step 3: Expand the right-hand side
Using the identity \((a - b)^2 = a^2 - 2ab + b^2\), we expand the right-hand side:

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

This simplifies to:

\[ (1 + \sin A) + (1 - \sin A) - 2 \sqrt{(1 + \sin A)(1 - \sin A)} \]

### Step 4: Combine like terms
Combining the terms gives:

\[ 2 - 2 \sqrt{1 - \sin^2 A} \]

Using the Pythagorean identity \(1 - \sin^2 A = \cos^2 A\), we can rewrite this as:

\[ 2 - 2 \cos A \]

### Step 5: Set the equation
Now we have:

\[ 4 \cos^2 \left( \frac{A}{2} \right) = 2 - 2 \cos A \]

### Step 6: Divide by 2
Dividing both sides by 2 gives:

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

### Step 7: Use the double angle formula
Recall the double angle formula for cosine:

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

Substituting this into our equation gives:

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

### Step 8: Simplify the equation
This simplifies to:

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

### Step 9: Combine like terms
Adding \(2 \cos^2 \left( \frac{A}{2} \right)\) to both sides gives:

\[ 4 \cos^2 \left( \frac{A}{2} \right) = 2 \]

### Step 10: Solve for \(\cos^2 \left( \frac{A}{2} \right)\)
Dividing both sides by 4 results in:

\[ \cos^2 \left( \frac{A}{2} \right) = \frac{1}{2} \]

### Step 11: Find \(\cos \left( \frac{A}{2} \right)\)
Taking the square root gives:

\[ \cos \left( \frac{A}{2} \right) = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2} \]

### Step 12: Determine possible angles
The angles for which \(\cos \left( \frac{A}{2} \right) = \frac{\sqrt{2}}{2}\) are:

\[ \frac{A}{2} = 45^\circ + n \cdot 360^\circ \quad \text{or} \quad \frac{A}{2} = 315^\circ + n \cdot 360^\circ \]

Thus, multiplying by 2 gives:

\[ A = 90^\circ + n \cdot 720^\circ \quad \text{or} \quad A = 630^\circ + n \cdot 720^\circ \]