Let's expand the left side of the equation:
(x-y)^2 = x^2 - 2xy + y^2(x+y)^2 = x^2 + 2xy + y^2
Adding these two expressions together, we get:
(x^2 - 2xy + y^2) + (x^2 + 2xy + y^2) = 2x^2 + 2y^2 = 2(x^2 + y^2)
Therefore, (x-y)^2 + (x+y)^2 = 2(x^2 + y^2)
Let's expand the left side of the equation:
(x-y)^2 = x^2 - 2xy + y^2
(x+y)^2 = x^2 + 2xy + y^2
Adding these two expressions together, we get:
(x^2 - 2xy + y^2) + (x^2 + 2xy + y^2) = 2x^2 + 2y^2 = 2(x^2 + y^2)
Therefore, (x-y)^2 + (x+y)^2 = 2(x^2 + y^2)