Using the steps below, find the value of $x ^ { 2012 } + x ^ { - 2012 }$, where $x + x ^ { - 1 } = \frac { \sqrt { 5 } + 1 } { 2 }$. a) For any real $r$, show that $\left| r + r ^ { - 1 } \right| \geq 2$. What does this tell you about the given $x$? b) Show that $\cos \left( \frac { \pi } { 5 } \right) = \frac { \sqrt { 5 } + 1 } { 4 }$, e.g. compare $\sin \left( \frac { 2 \pi } { 5 } \right)$ and $\sin \left( \frac { 3 \pi } { 5 } \right)$. c) Combine conclusions of parts a and b to express $x$ and therefore the desired quantity in a suitable form.
Using the steps below, find the value of $x ^ { 2012 } + x ^ { - 2012 }$, where $x + x ^ { - 1 } = \frac { \sqrt { 5 } + 1 } { 2 }$.\\
a) For any real $r$, show that $\left| r + r ^ { - 1 } \right| \geq 2$. What does this tell you about the given $x$?\\
b) Show that $\cos \left( \frac { \pi } { 5 } \right) = \frac { \sqrt { 5 } + 1 } { 4 }$, e.g. compare $\sin \left( \frac { 2 \pi } { 5 } \right)$ and $\sin \left( \frac { 3 \pi } { 5 } \right)$.\\
c) Combine conclusions of parts a and b to express $x$ and therefore the desired quantity in a suitable form.