TITLE: What is the value of $\arctan(1/2)+\arctan(1/5)+\arctan(1/8)$?
QUESTION [8 upvotes]: What is the value of :
  $$\arctan(1/2)+\arctan(1/5)+\arctan(1/8)?$$

I tried to do geometric solution:: 
Where in the angles we are looking for are shown, but I can't solve it.  Can we use it with this kind of approach? Can someone also post a solution using trigonometric identities?

REPLY [15 votes]: By considering that
$$(2+i)(5+i)(8+i) = 65(1+i)$$
and by taking the argument of both sides we immediately have
$$ \arctan\frac{1}{2}+\arctan\frac{1}{5}+\arctan\frac{1}{8}=\arctan 1=\frac{\pi}{4}.$$