Question

If $S=\sum _{r=1}^{90}ta{n}^{-1}\left(\frac{2r}{2+r^2+r^4}\right)$, then $8190cotS$ is equal to

Answer 1

${\tan }^{-1}(\frac{2r}{2+r^2+r^4})={\tan }^{-1}(\frac{(r^2+1+r)-(r^2+1-r)}{(r^2+1+r)(r^2+1-r)+1})={\tan }^{-1}(r^2+1+r)-{\tan }^{-1}(r^2+1-r)$
$S={\sum }_{r=1}^{90}\left[{\tan }^{-1}\right(r^2+1+r)-{\tan }^{-1}(r^2+1-r\left)\right]={\tan }^{-1}\left(8191\right)-{\tan }^{-1}\left(1\right)={\tan }^{-1}\left(\frac{8190}{8192}\right)$
$\Rightarrow 8190\cot S=8192$