Find `"sin"2theta` when `"sin"theta+costheta`=1.

Answer

To find the value of \(\sin 2\theta\) given that \(\sin \theta + \cos \theta = 1\), we can follow these steps:

### Step-by-Step Solution:

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

2. **Square both sides of the equation:**
\[ (\sin \theta + \cos \theta)^2 = 1^2 \]
This expands to:
\[ \sin^2 \theta + 2\sin \theta \cos \theta + \cos^2 \theta = 1 \]

3. **Use the Pythagorean identity:**
We know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
So we can substitute this into our equation:
\[ 1 + 2\sin \theta \cos \theta = 1 \]

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

5. **Solve for \(\sin 2\theta\):**
Recall that:
\[ \sin 2\theta = 2\sin \theta \cos \theta \]
Therefore, substituting from the previous step:
\[ \sin 2\theta = 0 \]

### Final Answer:
\[ \sin 2\theta = 0 \]