If sinx + cosx = a, then the value of| sinx - cosx| is

Options

A. `sqrt(2-a^(2)`
B. `sqrt(a^(2)-2)`
C. ` sqrt(a^(2)+2)`
D. 1

Answer

To find the value of |sin x - cos x| given that sin x + cos x = a, we can follow these steps:

### Step 1: Square the equation
We start with the equation:
\[ \sin x + \cos x = a \]
Now, we square both sides:
\[ (\sin x + \cos x)^2 = a^2 \]

### Step 2: Expand the left-hand side
Using the identity \((a + b)^2 = a^2 + b^2 + 2ab\), we expand:
\[ \sin^2 x + \cos^2 x + 2\sin x \cos x = a^2 \]

### Step 3: Use the Pythagorean identity
We know that:
\[ \sin^2 x + \cos^2 x = 1 \]
Substituting this into our equation gives:
\[ 1 + 2\sin x \cos x = a^2 \]

### Step 4: Solve for \(2\sin x \cos x\)
Rearranging the equation, we find:
\[ 2\sin x \cos x = a^2 - 1 \]

### Step 5: Square the expression for \(|\sin x - \cos x|\)
Next, we need to find \(|\sin x - \cos x|\). We can square this expression:
\[ |\sin x - \cos x|^2 = (\sin x - \cos x)^2 \]
Expanding this using the identity \((a - b)^2 = a^2 + b^2 - 2ab\):
\[ \sin^2 x + \cos^2 x - 2\sin x \cos x \]

### Step 6: Substitute known values
Using \(\sin^2 x + \cos^2 x = 1\) again, we have:
\[ |\sin x - \cos x|^2 = 1 - 2\sin x \cos x \]
Now, substitute \(2\sin x \cos x\) from Step 4:
\[ |\sin x - \cos x|^2 = 1 - (a^2 - 1) \]

### Step 7: Simplify the expression
This simplifies to:
\[ |\sin x - \cos x|^2 = 1 - a^2 + 1 = 2 - a^2 \]

### Step 8: Take the square root
Finally, to find \(|\sin x - \cos x|\), we take the square root:
\[ |\sin x - \cos x| = \sqrt{2 - a^2} \]

Thus, the value of \(|\sin x - \cos x|\) is:
\[ \boxed{\sqrt{2 - a^2}} \]