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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