Summation related to arctan

Question

What is $$\sum _{r=1} ^{90} \arctan \left(\frac{2r}{2+r^2+r^4}\right)$$ . Now I tried to express it as sum of $\arctan(a)-\arctan(b)$ . My try was using $r+\frac{1}{r}=a,r-\frac{1}{r}=b$ everything worked well except the term independent of variable. .all the problem is around that 2 in denominator. How do I tackle it?

Answer 1

Hint:

$$\frac{2r}{2+r^2+r^4} = \frac{(r^2+r+1)-(r^2-r+1)}{1+(r^2+r+1)(r^2-r+1)}$$

Now you may proceed!