Show that if $x + \frac1x = 1$, then $x^5 + \frac1{x^5} = 1$. Suppose $x + \frac{1}{x} = 1$.
Without first working out what $x$ is, show that 
$x^5 + \frac{1}{x^5} = 1$ as well.

Put $u=x+\frac 1x$
and let 
$u^\boxed{\tiny n}=x^n+\frac 1{x^n}$.
By expansion it is clear that
$$u^3=u^\boxed{\tiny{3}}+3u$$
and
$$u^5=u^\boxed{\tiny5}+5u^\boxed{\tiny3}+10u$$.
Eliminating $u^\boxed{\tiny3}$ gives
$$u^{\boxed{\tiny5}}=u^5-5u^3+5u=1-5+5=1$$
