+499 since i haven't seen anyone answer yet, i'll try pitching my 2 cents.
we have that z+1/z = 1
if we square our given expression, we get:
z^2 + 2 + 1/z^2 = 1
z^2 + 1/z^2 = -1
if we square this again, we get:
z^4 + 2 + 1/z^4 = 1
z^4 + 1/z^4 = -1
as you might be able to deduce, this will equal -1 however many times we square our expression, because the 2ab term will always equal 2, and the RHS(right hand side) will always equal (-1)^2 = 1, so subtract the two, and you get -1.
z^2 + 1/z^2 = -1
z^4 + 1/z^4 = -1
z^8 + 1/z^8 = -1
Now we have a problem because we are asked to find z^10 + 1/z^10 which does not entail these powers of 2. We can try experimenting to get the z^10 term, and the first thing that comes to mind is multiplying by z^2 + 1/z^2
(z^2+1/z^2)(z^8+1/z^8) = z^10 + 1/z^10 + z^6 + 1/z^6 = 1
now we need the value of z^6 + 1/z^6 to get the value of our desired expression.
To do so, we can repeat these steps with z^2+1/z^2 and z^4 + 1/z^4
(z^2 + 1/z^2)(z^4+1/z^4) = z^6 + 1/z^6 + z^2 + 1/z^2 = 1
this is good, because we have the value of z^2 + 1/z^2 = -1
z^6 + 1/z^6 - 1 = 1
z^6 + 1/z^6 = 2
now we go back to our expression with the z^10 term and substitute the value of z^6 + 1/z^6 to get;
z^10 + 1/z^10 + 2 = 1
z^10 + 1/z^10 = -1
