(z + 1/z)^2 = 1
z^2 + 2z (1/z) + 1/z^2 = 1
z^2 + 2 + 1/z^2 = 1 subtract 2 from both sides
z^2 + 1/z^2 = -1
(z^2 + 1/z^2)^2 = 1
z^4 + 2 z^2(1/z^2) + 1/z^4 = 1
z^4 + 2 + 1/z^4 = 1 subtract 2 from both sides
z^4 + 1/z^4 = -1
(z^4 + 1/z^4)^3 = ( -1 )^3 binomial expansion on the left side
(z^4)^3 + 3(z^4)^2(1/z^4) + 3(z^4)(1/z^4)^2 + (1/z^4)^3 = -1
z^12 + 3(z^8/z^4) + 3(z^4/z^8 + 1/z^12 = -1
z^12 + 3z^4 + 3/z^4 + 1/z^12 = -1
z^12 + 3 (z^4 + 1/z^4) + 1/z^12 = -1
z^12 + 3(-1) +1/z^12 = -1
z^12 + 1/z^12 - 3 = -1 add 3 to both sides
z^12 + 1/z^12 = 2