(1+i)^2016 =((1+i)^16)^126 =256^126
(1 - i)^2016=((1-i)^16)^126 =256^126
256^126 - 256^126 =0
( 1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i
(1 - i)^2 = 1 - 2i + i^2 = 1 - 2i - 1 = -2i
So
( 1 + i)^2016 - (1 - i)^2016 =
[ (1 + i)^2 ] ^1008 - [ (1 - i)^2 ]^1008 =
(2i)^1008 - (-2i)^1008 -
(2i)^1008 - (-2)^1008 * (i)^1008 = [ (-2)^even power = 2^even power]
(2i)^1008 - (2)^1008 * i^1008 =
(2i)^1008 - (2i)^1008 =
0