The value of (sin A - cos A)² + (sin A + cos A)² is

The given expression is (sin A - cos A)² + (sin A + cos A)².

Let's simplify it step by step:

Step 1: Expand the squares

(sin A - cos A)² = sin² A - 2sin A cos A + cos² A (sin A + cos A)² = sin² A + 2sin A cos A + cos² A

Step 2: Add the two expanded expressions

(sin² A - 2sin A cos A + cos² A) + (sin² A + 2sin A cos A + cos² A)

Step 3: Simplify the expression

2sin² A + 2cos² A = 2(sin² A + cos² A)

Step 4: Use the Pythagorean identity sin² A + cos² A = 1

2(1) = 2

So, the value of (sin A - cos A)² + (sin A + cos A)² is 2.