<s>[INST] Find the exact value of $\cos\frac{2\pi}{5}$ by solving equation There's this concept in the topic of complex numbers which I don't really understand much (and was devastated upon realising it'll appear in the topic often) - trigonometry!
I'm super lost to be honest.
We are asked to express $\cos3\theta$ and $\cos2\theta$ in terms of $\cos\theta$. 
Then show that $\cos3\theta=\cos2\theta$ can be written as $4z^3-2z^2-3z+1=0$ where $z=\cos\theta$.
Lastly, by solving the equation above for $z$, we are to find out the value of $\cos\frac{2\pi}{5}$.
I attempted the question with just subbing in $z=\cos\theta$ into the equation and I got: $$4\cos^3\theta-3\cos\theta=\cos2\theta$$ and don't really know where to go next.
 [/INST] We have 
\begin{eqnarray*}
\cos( 2 \theta) =2 \cos^2(\theta)-1 \\
\cos( 4 \theta) =4 \cos^3(\theta)-3 \cos(  \theta)
\end{eqnarray*}
Let $z=\cos( \theta) $ then $\cos(3 \theta) = \cos(2 \theta)$ becomes $4z^3-2z^2-3z+1=0$. This will factorise $(z-1)(4z^2+2z-1)=0$, so $\color{red}{\cos(\frac{2 \pi}{5} )=\frac{-1+\sqrt{5}}{4}}$.
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