Applying the identities
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
(i) (a + b)2 – (a – b)2 = a2 + 2ab + b2 – [a2 – 2ab + b2]
= a2 + 2ab + b2 – a2 + 2ab – b2
= a2 (1 – 1) + ab (2 + 2) + b2 (1 – 1)
= 0a2 + 4 ab + 0b2
= 4ab
(a + b)2 – (a – b)2 = 4ab
(ii) (a + b)2 + (a – b)2 = a2 + 2ab + b2 + (a2 – 2ab + b2)
= a2 + 2ab + b2 + a2 – 2ab + b2
= a2 (1 + 1) + ab (2 – 2) + b2 (1 + 1)
= 2a2 + 0 ab + 2b2
= 2a2 + 2b2
= 2(a2 + b2)
∴ (a + b)2 – (a – b)2 = 2(a2 + b2)
