Question

If \( \sin A + \cos A = \sqrt{2} \), find the value of \( \sin A \cos A \).

Hint:

Use the identity \( (\sin A + \cos A)^2 = 1 + 2 \sin A \cos A \) to solve problems involving \( \sin A + \cos A \).

Answer 1

Step 1: Use the identity for square of sine and cosine.
We know the identity: \[ (\sin A + \cos A)^2 = \sin^2 A + \cos^2 A + 2 \sin A \cos A \]
Step 2: Substitute the given value.
It is given that \( \sin A + \cos A = \sqrt{2} \). Squaring both sides, we get: \[ (\sqrt{2})^2 = \sin^2 A + \cos^2 A + 2 \sin A \cos A \] \[ 2 = 1 + 2 \sin A \cos A \] (Since \( \sin^2 A + \cos^2 A = 1 \))
Step 3: Solve for \( \sin A \cos A \).
\[ 2 \sin A \cos A = 2 - 1 = 1 \] \[ \sin A \cos A = \frac{1}{2} \]