Most elegant proof of Arctan(x) + Arctan(1/x)
You have a right triangle. What do the two non 90 degree angles add up to? Oh yeah, that's right, 90 degrees. 90 degrees obviously = pi/2 radians. What more do you need? Arctan(x) finds one of the angles, Arctan(1/x) finds the other.
EDIT: Just realized mathwonk said the same thing.