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

Options

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

Answer

To solve the equation \( \sin \theta + \cos \theta = 1 \) and 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:
\[ (\sin \theta + \cos \theta)^2 = 1^2 \]

### Step 2: Expand the left-hand side
Using the identity \( (A + B)^2 = A^2 + B^2 + 2AB \), we can expand the left-hand side:
\[ \sin^2 \theta + \cos^2 \theta + 2\sin \theta \cos \theta = 1 \]

### Step 3: Use the Pythagorean identity
We know from trigonometric identities that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting this into our equation gives:
\[ 1 + 2\sin \theta \cos \theta = 1 \]

### Step 4: Simplify the equation
Now, we can simplify the equation:
\[ 2\sin \theta \cos \theta = 1 - 1 \]
\[ 2\sin \theta \cos \theta = 0 \]

### Step 5: Find \( \sin 2\theta \)
We know that:
\[ \sin 2\theta = 2\sin \theta \cos \theta \]
From the previous step, we have:
\[ \sin 2\theta = 0 \]

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