If `sintheta+costheta=1`, then the value of ` sin2theta` is

Options

A. 1
B. `(1)/(2)`
C. 0
D. -1

Explanation

To solve the problem where \( \sin \theta + \cos \theta = 1 \) and we need to find the value of \( \sin 2\theta \), we can follow these steps:

### Step 1: Square both sides of the equation
We start with the equation:
\[ \sin \theta + \cos \theta = 1 \]
Now, we square both sides:
\[ (\sin \theta + \cos \theta)^2 = 1^2 \]
This simplifies to:
\[ \sin^2 \theta + \cos^2 \theta + 2\sin \theta \cos \theta = 1 \]

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

### Step 3: Simplify the equation
Now, we can simplify the equation:
\[ 1 + 2\sin \theta \cos \theta = 1 \]
Subtracting 1 from both sides results in:
\[ 2\sin \theta \cos \theta = 0 \]

### Step 4: Relate to \( \sin 2\theta \)
We know that:
\[ \sin 2\theta = 2\sin \theta \cos \theta \]
Thus, from our previous result, we have:
\[ \sin 2\theta = 0 \]

### Conclusion
The value of \( \sin 2\theta \) is:
\[ \sin 2\theta = 0 \]