How to obtain $\sqrt{5}$ (without a calculator) using complex numbers & roots of unity? For context, here is the entire question:

15. (a) Find the fifth roots of unity in exponential form.$\hspace{14.5pt}$(b) Let $\alpha$ be the complex fifth root of unity with the smallest positive argument, and suppose that $u = \alpha + \alpha^4$ and $v = \alpha^2 + \alpha^3$,$\hspace{30pt}$(i) Find the values of $u + v$ and $u - v$.

Particularly, I am struggling with the second part of (b)(i) where the answer is $\sqrt{5}$ for $u-v$. However, I cannot fathom how this answer is obtained without using a calculator.
What I have done so far: $u-v = \alpha + \alpha^4 - \alpha^2 - \alpha^3$, then through substitution of $\alpha = e^{i(2\pi/5)}$, I eventually obtained $u - v = 2\cos(2\pi/5) - 2\cos(4\pi/5)$.
How do I go further from here to calculate √5, or is it a distinct approach to the question entirely?
Any help greatly appreciated!

$z^5 - 1 = 0\\
(z-1)(z^4+z^3+z^2 + z+1)=0$
$z^4+z^3+z^2 +z+ 1$ might not be immediately obvious how to factor, but it is a symmetric polynomial so we can do this.
$z^2(z^2 + \frac 1{z^2} + z+ \frac {1}{z} + 1)$
let $s =  z+ \frac 1z$
$s^2 = z^2 + 2 + \frac {1}{z^2}$
$s^2 + s - 1 = 0$
$s = \frac {-1\pm\sqrt{5}}{2}$
There is a shortcut from here to the end, and the next few lines can be bypassed.
$z + \frac 1z = \frac {-1\pm\sqrt{5}}{2} $
$z^2 + \frac {1 \pm \sqrt 5}{2}z + 1 = 0$
$z = \frac {-1\pm\sqrt{5}}{4} \pm \sqrt{\frac {5 \pm \sqrt 5}{8}}i$
or $z = \cos(\frac {2n\pi}{5})+i\sin(\frac {2n\pi}{5})$
The root with the smallest argument is $a = \frac {-1 + \sqrt{5}}{4} +  \sqrt{\frac {5 + \sqrt 5}{8}}i = \cos(\frac {2\pi}{5})+i\sin(\frac {2\pi}{5}) = e^{\frac {2\pi}5 i} $
$u = a+a^4 = e^{\frac {2\pi}5 i}+ e^{\frac {8\pi}5 i} = e^{\frac {2\pi}5 i}+e^{\frac {-2\pi}5 i} = 2\text{ Re} (a) = \frac{-1+\sqrt 5}{2}$
$v= a^2+a^3 = e^{\frac {4\pi}5 i}+ e^{\frac {6\pi}5 i} = e^{\frac {4\pi}5 i}+e^{\frac {-4\pi}5 i} = 2\text{ Re} (a^2) = \frac {-1-\sqrt 5}{2}$
Since we only needed the real part of $a$ we could save ourselves a trip through the quadratic formula to find the imaginary part of $a$
$u+v = -1$
$u-v = \sqrt 5$
