Question

If sin θ + cos θ = √2, what is the value of sin θ · cos θ?

• A. 1/4
• B. 1/2
• C. 1/√2
• D. 1

Hint:

Square trigonometric sums to introduce \( \sin \theta \cos \theta \) terms, leveraging identities like \( \sin^2 \theta + \cos^2 \theta = 1 \).

Answer 1

The Correct Option is B

To find \( \sin \theta \cos \theta \), we follow these steps:

1. Given: \( \sin \theta + \cos \theta = \sqrt{2} \).
2. Square both sides to use trigonometric identities: \[ (\sin \theta + \cos \theta)^2 = (\sqrt{2})^2. \]
3. Expand the left-hand side: \[ \sin^2 \theta + \cos^2 \theta + 2 \sin \theta \cos \theta = 2. \]
4. Use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \): \[ 1 + 2 \sin \theta \cos \theta = 2. \]
5. Solve for \( \sin \theta \cos \theta \): \[ 2 \sin \theta \cos \theta = 2 - 1 = 1 \implies \sin \theta \cos \theta = \frac{1}{2}. \]
6. Match with options: \( \frac{1}{2} \) corresponds to option (B).

Thus, the correct answer is: \[ \boxed{\frac{1}{2}} \]