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-by-Step Solution:

1. **Start with the given equation:**
\[ \sin \theta + \cos \theta = 1 \]

2. **Square both sides:**
\[ (\sin \theta + \cos \theta)^2 = 1^2 \]
This gives:
\[ \sin^2 \theta + \cos^2 \theta + 2\sin \theta \cos \theta = 1 \]

3. **Use the Pythagorean identity:**
We know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting this into the equation from step 2, we get:
\[ 1 + 2\sin \theta \cos \theta = 1 \]

4. **Simplify the equation:**
Subtract 1 from both sides:
\[ 2\sin \theta \cos \theta = 0 \]

5. **Use the double angle identity:**
We know that:
\[ \sin 2\theta = 2\sin \theta \cos \theta \]
Therefore, substituting this into our equation gives:
\[ \sin 2\theta = 0 \]

6. **Conclusion:**
Thus, the value of \( \sin 2\theta \) is:
\[ \sin 2\theta = 0 \]

### Final Answer:
The value of \( \sin 2\theta \) is \( 0 \).

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