`(1-sinA+cosA)^2` is equal to :

Options

A. `2(1-cosA)(1+sinA)`
B. 2(1-sinA)(1+cosA)
C. 2(1-cosA)(1-sinA)
D. None of these

Answer

To simplify the expression \((1 - \sin A + \cos A)^2\), we can follow these steps:

### Step 1: Expand the expression
We start with the expression:
\[ (1 - \sin A + \cos A)^2 \]
Using the formula for the square of a binomial, \((a + b)^2 = a^2 + 2ab + b^2\), we can rewrite it as:
\[ (1 - \sin A + \cos A)(1 - \sin A + \cos A) \]

### Step 2: Apply the expansion
Now, we will expand this expression:
\[ = (1 - \sin A)^2 + 2(1 - \sin A)(\cos A) + (\cos A)^2 \]

### Step 3: Calculate \((1 - \sin A)^2\)
Calculating \((1 - \sin A)^2\):
\[ (1 - \sin A)^2 = 1 - 2\sin A + \sin^2 A \]

### Step 4: Calculate \((\cos A)^2\)
We know that:
\[ (\cos A)^2 = \cos^2 A \]

### Step 5: Combine the results
Now we can combine all the parts:
\[ 1 - 2\sin A + \sin^2 A + 2(1 - \sin A)(\cos A) + \cos^2 A \]

### Step 6: Substitute \(\sin^2 A + \cos^2 A = 1\)
Using the Pythagorean identity \(\sin^2 A + \cos^2 A = 1\):
\[ = 1 - 2\sin A + 1 + 2(1 - \sin A)(\cos A) \]
\[ = 2 - 2\sin A + 2(1 - \sin A)(\cos A) \]

### Step 7: Factor out common terms
Now, we can factor out \(2\):
\[ = 2(1 - \sin A + (1 - \sin A)\cos A) \]
\[ = 2(1 - \sin A)(1 + \cos A) \]

### Final Result
Thus, the expression \((1 - \sin A + \cos A)^2\) simplifies to:
\[ 2(1 - \sin A)(1 + \cos A) \]