How to do that complex number question

W=e^(2I(pi)/5)
U=w+w^4
V=w^2+w^3
Show that u+v=-1
uv=-1
u-v=root 5
Hence find the exact value for cos 2pi/5
Thanks

First, you need to make use of the Euler formula:
e^(ix) = cos x + i sin x

and thus recognize that w=e^(2iπ/5) is the principle 5th root of 1, and so
w^k = e^(2kiπ/5), k = 0,...,4
are all five 5th roots of 1, in cyclic order

Second, it is often very enlightening to draw the Argand diagram (complex plane plot) of things like this. It will reveal lots of relations about these quantities that will be very useful.

But this one can be done entirely with the algebraic approach.
Note that
w⁵ = 1
and that the powers of w, from 0 to 4, are the 5 roots of
z⁵ = 1,
z⁵ - 1 = 0, which factors:
(z - 1)(z⁴ + z³ + z² + z + 1) = 0
so those roots are
z = 1, and the 4 roots of
z⁴ + z³ + z² + z + 1 = 0, for which
z⁴ + z³ + z² + z = -1

u & v are defined by:
u = w + w⁴ = w + w*
v = w² + w³ = w² + (w²)*
Note that this makes both u and v real.

So u + v = w⁴ + w³ + w² + w = -1, by that quartic polynomial in z, just above.

And then you have
uv = (w + w⁴)(w² + w³) = w⁷ + w⁶ + w⁴ + w³
But because w⁵ = 1, you can reduce, by 5, any powers of w that are > 5:
w⁷ + w⁶ + w⁴ + w³ = w² + w + w⁴ + w³
But that's just what we got for u+v, rearranged! So
uv = -1

Next is u-v, for which, just use
(u - v)² = u² - 2uv + v²
= (u² + 2uv + v²) - 4uv
= (u + v)² - 4uv
= 1 + 4 = 5

So u - v = √5 [not -√5, because u>0 and v<0, so u-v>0]

Now invoke the Euler relation to find that
w = e^(2iπ/5) = cos(2π/5) + i sin(2π/5)
w* = e^(-2iπ/5) = cos(2π/5) - i sin(2π/5)
u = w + w* = 2cos(2π/5)
But
u = [(u+v) + (u-v)]/2 = (-1 + √5)/2
So
cos(2π/5) = (-1 + √5)/4

And thanks to you, for the BA!

And for posting this in the first place. I found it intriguing, because in the past, I have arrived at that result by other means, not involving complex numbers.