If `sin theta + cos theta=1,` what is the value of `sin theta cos theta`?

Options

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

Answer

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

### Step 1: Start with the given equation
We have:
\[ \sin \theta + \cos \theta = 1 \]

### Step 2: Square both sides
Squaring both sides of the equation gives:
\[ (\sin \theta + \cos \theta)^2 = 1^2 \]
This expands to:
\[ \sin^2 \theta + 2 \sin \theta \cos \theta + \cos^2 \theta = 1 \]

### Step 3: 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 4: Simplify the equation
Now we can simplify the equation:
\[ 2 \sin \theta \cos \theta = 1 - 1 \]
This simplifies to:
\[ 2 \sin \theta \cos \theta = 0 \]

### Step 5: Solve for \( \sin \theta \cos \theta \)
Dividing both sides by 2 gives:
\[ \sin \theta \cos \theta = 0 \]

### Conclusion
Thus, the value of \( \sin \theta \cos \theta \) is:
\[ \boxed{0} \]

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