Extension Complex Locus (1 Viewer)

[nichorowitz]

The question is describe the locus of Im (z+1/z) = 0
i.e.
http://www4c.wolframalpha.com/Calcu...bhh358f?MSPStoreType=image/gif&s=33&w=94&h=36

Now, the obvious method is to let z + x+iy and after a couple of lines of working you arrive at the answer
y(x^2 + y^2 - 1) = 0
http://www4c.wolframalpha.com/Calcu...a974ee?MSPStoreType=image/gif&s=37&w=120&h=22

Giving you the graph of the unit circle centred at the origin.
and the line y=0

simple right?

I then thought about this question in a different manner and thought about using the result
z+1/z = 2cos(Ɵ)

Proof goes like this:
let z = rcisƟ
and then by De Moivre's Theorem
z + 1/z = z + z^-1
= cisƟ + cis(-Ɵ)
= cosƟ + isinƟ + cosƟ - isinƟ
= 2cosƟ
therefore Im(z + 1/z) = 0
now in regards to question I apparently am supposed to describe the locus of the equation 0 = 0?

even wolfram alpha gives the odd locus.
http://www.wolframalpha.com/input/?i=Im(z+1/z)+=+0

Just upon typing this i figured out what was wrong with my working. Still gonna post anyway to see if you guys can figure it out as well.

Have fun.