# Simplifying $\arctan\frac{2}{1^2} + \arctan\frac{2}{2^2} + \arctan\frac{2}{3^2}+ \cdots+\arctan\frac{2}{n^2}$

$$S = \arctan\frac{2}{1^2} + \arctan\frac{2}{2^2} + \arctan\frac{2}{3^2}+ \cdots+\arctan\frac{2}{n^2}$$

I tried to simplify this summation by substituting $$\arctan(2/k^2)$$ to $$a_k$$ so that$$\ \tan(a_k) = 2/k^2$$, then using the sum of tangent formula, but it didn't work. Can someone help me how to simplify this summation?

$$\dfrac2{n^2}=\dfrac{n+1-(n-1)}{1+(n-1)(n+1)}$$
• You should add the not-well-known-formula-by-present-day-students $\tan(a-b)=\dfrac{\tan a - \tan b}{1+\tan a \tan b}$ Mar 11 '20 at 11:05
• @lab Bhattacharjee Thank you, sir... What a unique IDEA!! Thank you for your generosity :) I think I will always transform $2/n^2$ formula afterwards :)) Mar 11 '20 at 11:16