If `sintheta+costheta=sqrt(2)`, then find the value of `sintheta-costheta`.

Options

A. `sqrt(2)`
B. 0
C. `2sqrt(2)`
D. 2

Answer

To solve the problem where \( \sin \theta + \cos \theta = \sqrt{2} \) and we need to find the value of \( \sin \theta - \cos \theta \), we can follow these steps:

### Step 1: Set up the equations
Let:
1. \( \sin \theta + \cos \theta = \sqrt{2} \) (Equation 1)
2. \( \sin \theta - \cos \theta = x \) (Equation 2)

### Step 2: Square both equations
Now, we will square both equations.

From Equation 1:
\[ (\sin \theta + \cos \theta)^2 = (\sqrt{2})^2 \]
This expands to:
\[ \sin^2 \theta + 2\sin \theta \cos \theta + \cos^2 \theta = 2 \]
Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can substitute:
\[ 1 + 2\sin \theta \cos \theta = 2 \]
This simplifies to:
\[ 2\sin \theta \cos \theta = 1 \]
Thus:
\[ \sin \theta \cos \theta = \frac{1}{2} \]

From Equation 2:
\[ (\sin \theta - \cos \theta)^2 = x^2 \]
This expands to:
\[ \sin^2 \theta - 2\sin \theta \cos \theta + \cos^2 \theta = x^2 \]
Again, using the Pythagorean identity:
\[ 1 - 2\sin \theta \cos \theta = x^2 \]
Substituting \( \sin \theta \cos \theta = \frac{1}{2} \):
\[ 1 - 2 \cdot \frac{1}{2} = x^2 \]
This simplifies to:
\[ 1 - 1 = x^2 \]
Thus:
\[ x^2 = 0 \]

### Step 3: Solve for \( x \)
Taking the square root of both sides:
\[ x = 0 \]
Since we defined \( x = \sin \theta - \cos \theta \), we have:
\[ \sin \theta - \cos \theta = 0 \]

### Conclusion
Therefore, the value of \( \sin \theta - \cos \theta \) is \( 0 \).