If z is a complete number satisfying
I assume that you mean complex number rather than complete number.
There are two obvious methods, first, you can build upto the result by a series of multiplications and squarings.
So for starters (z+1/z)^2 = z^2 + 2 + 1/z^2 = 1, so z^2 + 1/z^2 = -1.
Squaring that, z^4 + 2 + 1/z^4 = 1, so z^4 + 1/z^4 = -1 and so on.
The intermediate expression z^3 + 1/z^3 can be obtained by multiplying z + 1/z by z^2 + 1/z^2 etc..
A second method requires you to solve the equation z + 1/z = 1.
That gets you z = 1/2 +- i*sqrt(3)/2 = cos(pi/3) +- i*sin(pi/3) and now you can use De'Moivre's theorem.