This is basic trig, you don't need to learn "more" trig, you just need to learn basic trig... it may help if I tell you that
arctan is just another way of writing
tan−1. Also,
tan2π=1, so I'm not sure why you "know" it. In fact, the tangent of
2π is undefined, so that's literally one of the worst things you could have claimed to be equal to 1, since it's about as far off being equal to 1 as one can possible even get.
Anyways, to address your question: look at an argand diagram, you want the argument of a complex number to be pi/4, this means it needs to be a line that angled at pi/4 to the (positive) real axis and pi/4 to the (positive) imaginary axis. This is precisely (if you sketch the line) the line
y=x, i.e:
ℜ(z)=ℑ(z).
Trig is in radians... obviously. How does "this is radians" imply that it isn't trig? That's as contradictory as you get.