Key idea. To prove expressions equal, factor their difference.

1. Use \(z=-(x+y)\) in the sum of squares:

   \[2=x^2+y^2+(x+y)^2=2(x^2+xy+y^2).\]

   Hence \(x^2+xy+y^2=1\).

2. Subtract two of the expressions:

   \[\begin{aligned}(x^3-x)-(y^3-y)&=(x^3-y^3)-(x-y)\\&=(x-y)(x^2+xy+y^2-1)\\&=0.\end{aligned}\]

   Thus \(x^3-x=y^3-y\).

3. The hypotheses treat x, y and z in the same way. Repeating the argument with y and z gives \(y^3-y=z^3-z\). All three expressions are therefore equal.

Remember. Equality can be easier to prove by showing that a difference is zero.