${\mathrm{Cot}}^{-1}3+{\mathrm{Cot}}^{-1}7+{\mathrm{Cos}}^{-1}13+···n terms$ =

$$\begin{gathered} {\tan }^{-1}\left(\frac{1}{3}\right)+{\tan }^{-1}\left(\frac{1}{7}\right)+{\tan }^{-1}\left(\frac{1}{13}\right)... \\ {\tan }^{-1}\left(\frac{2-1}{1+2·1}\right)+{\tan }^{-1}\left(\frac{3-2}{1+3·2}\right)+{\tan }^{-1}\left(\frac{4-3}{1+4·3}\right)... \\ =\left({\tan }^{-1}2-{\tan }^{-1}1\right)+\left({\tan }^{-1}3-{\tan }^{-1}2\right).....+{\tan }^{-1}\left(\mathrm{n}+1\right)-{\tan }^{-1}\mathrm{n} \\ ={\tan }^{-1}\left(\mathrm{n}+1\right)-{\tan }^{-1}1={\tan }^{-1}\left(\frac{\mathrm{n}}{1+\mathrm{n}+1}\right) \\ ={\tan }^{-1}\left(\frac{\mathrm{n}}{\mathrm{n}+2}\right) \end{gathered}$$