If `sintheta + cos theta =1`, then the value of `sintheta - costheta` is:

Options

A. a)0
B. b)`+-sqrt(2)`
C. c)`+-1`
D. d)`+- 1/sqrt(2)`

Answer

To solve the equation \( \sin \theta + \cos \theta = 1 \) and find the value of \( \sin \theta - \cos \theta \), we can follow these steps:

### Step 1: Square both sides of the equation
Starting with the equation:
\[ \sin \theta + \cos \theta = 1 \]
Square both sides:
\[ (\sin \theta + \cos \theta)^2 = 1^2 \]

### Step 2: Expand the left side
Using the identity \( (a + b)^2 = a^2 + b^2 + 2ab \):
\[ \sin^2 \theta + \cos^2 \theta + 2 \sin \theta \cos \theta = 1 \]
We know from the Pythagorean identity that \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ 1 + 2 \sin \theta \cos \theta = 1 \]

### Step 3: Simplify the equation
Subtract 1 from both sides:
\[ 2 \sin \theta \cos \theta = 0 \]

### Step 4: Solve for \( \sin \theta \) and \( \cos \theta \)
This implies:
\[ \sin \theta \cos \theta = 0 \]
Thus, either \( \sin \theta = 0 \) or \( \cos \theta = 0 \).

### Step 5: Find possible values of \( \theta \)
1. If \( \sin \theta = 0 \), then \( \theta = 0^\circ, 180^\circ, \ldots \)
2. If \( \cos \theta = 0 \), then \( \theta = 90^\circ, 270^\circ, \ldots \)

### Step 6: Calculate \( \sin \theta - \cos \theta \)
Now we can evaluate \( \sin \theta - \cos \theta \) for the possible angles:
- For \( \theta = 0^\circ \):
\[ \sin 0^\circ - \cos 0^\circ = 0 - 1 = -1 \]
- For \( \theta = 90^\circ \):
\[ \sin 90^\circ - \cos 90^\circ = 1 - 0 = 1 \]

### Conclusion
The possible values of \( \sin \theta - \cos \theta \) are \( -1 \) and \( 1 \).

Thus, the final answer is:
\[ \sin \theta - \cos \theta = \pm 1 \]