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.

Since $x + \frac{1}{x} = 1$ then
\begin{align}
1 &= 1^5 = \left( x + \frac{1}{x} \right)^{5} \\
&= \left( x^5 + \frac{1}{x^5} \right) + 5 \, \left( x^3 + \frac{1}{x^3} \right) + 10 \, \left( x + \frac{1}{x} \right) \\
&= \left( x^5 + \frac{1}{x^5} \right) + 5 \, \left(x + \frac{1}{x} \right)^3 - 15 \, \left( x + \frac{1}{x} \right) + 10 \, \left( x + \frac{1}{x} \right) \\
1 &= \left( x^5 + \frac{1}{x^5} \right) + 5 \, \left(x + \frac{1}{x} \right)^3 - 5 \, \left( x + \frac{1}{x} \right) \\
1 &= x^5 + \frac{1}{x^5}
\end{align}
