We have to prove that (cospi/4+i*sinpi/4)^2008 is real.
cos (pi/4) = 1/sqrt 2
sin (pi/4) = 1/sqrt 2
(cos pi/4 + i*sin pi/4)^2008
=> [1/sqrt 2 + i*(1/sqrt 2)]^2008
=> [1/sqrt 2 + i*(1/sqrt 2)]^2^1004
=> [1/2 + i^2/2 + 2*(1/sqrt 2)(1/sqrt 2)]^1004
substitute i^2 = -1
=> [1/2 - 1/2 + 2*i*(1/sqrt 2)(1/sqrt 2)]^1004
=> [2*i*(1/2)]^1004
=> i^1004
=> i^2^502
=> (-1)^502
=> 1
This proves that (cos pi/4 + i*sin pi/4)^2008 = 1 is real.
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