let z=cosx+isinx. so zz*=|z|^2=1, z*=1/z, (z* conjugate)

z+z*=2cosx, z-z*=2isinx

(z+z*)^8+(z-z*)^8=2^8(17/2^5)
(z+z*)^8+(z-z*)^8=17*8
(z^2+z*^2+2)^4+(z^2+z*^2-2)^4=17*8
let t=z^2+z*^2=2cos(2x)

(t+2)^4+(t-2)^4=17*8
2*t^4+2*6*4*t^2+32=17*8
t^4+24t^2-52=0
(t^2-2)(t^2+26)=0

2cos(2x)=-26
cos(2x)=-13, but |cos(2x)|<=1 so no roots

cos(2x)^2=1/2
cos(2x)=+/- 1/sqrt(2)
so for some integer k
2x=pi/4+(k/2)pi
x=pi/8+kpi/4

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