Prove that the number (cospi/4+i*sinpi/4)^2008 is real.

We have to prove that (cos pi/4 + i*sin pi/4)^2008 is
real.

Now (cos pi/4 + i*sin
pi/4)^2008

=> (cos pi/4 + i*sin
pi/4)^2^1004

=> [ (cos pi/4)^2 + i^2*(sin pi/4)^2 +
2*i*(cos pi/4)(sin pi/4)]^1004

Now i^2 =
-1

=> [ (cos pi/4)^2 - (sin pi/4)^2 + 2*i*(cos
pi/4)(sin pi/4)]^1004

(cos x)^2 - (sin x)^2 = cos
2x

=> [(cos pi/2) + 2*i*(cos pi/4)(sin
pi/4)]^1004

cos (pi/2) =
0

=> [2*i*(cos pi/4)(sin
pi/4)]^2^502

=> [4*i^2 *(cos pi/4)^2*(sin
pi/4)^2]^502

=> [-16*(cos pi/4)^2*(sin
pi/4)^2]^502

This is real as i has been
eliminated.

Therefore (cos pi/4 + i*sin
pi/4)^2008 is real.