The value of cot ( Σ _ n = 1 ^ 23 cot ^ - 1 (1 + Σ _ k = 1 ^ n 2k))...

$\cot \left(\mathop{Σ}_{\text{n} = 1}^{23}{\text{cot}}^{- 1}\left(1 + \mathop{Σ}_{\text{k} = 1}^{\text{n}}2\text{k}\right)\right) = \text{cot}\mathop{Σ}_{\text{n} = 1}^{23}{\text{cot}}^{- 1}\left(1 + 2 + 4 + 6 + \ldots + 2\text{n}\right)$
\text{=cot}Σ{\text{cot}}^{- 1}\left(1 + \text{n}\left(\text{n} + 1\right)\right)$<br />$\text{=cot}Σ{\text\tan}^{- 1}\dfrac{\left(\text{n} + 1\right) - \text{n}}{1 + \text{n}\left(\text{n} + 1\right)}$\left(co{t}^{- 1}x = \tan^{- 1}\dfrac{1}{x}\right)$
\text{=cot}\mathop{Σ}_{\text{n} = 1}^{23}\left({\text\tan}^{- 1}\left(\text{n} + 1\right) - {\text\tan}^{- 1}\text{n}\right)\left[{\text\tan}^{\text{-1}}\left(\dfrac{\text{x-y}}{1 + xy}\right){\text{=tan}}^{\text{-1}}{\text{x-tan}}^{\text{-1}}\text{y}\right]

Now {\text\tan}^{- 1}\left(\text{n} + 1\right) - {\text\tan}^{- 1}{\text{n=tan}}^{- 1}\left(2\right) - {\text\tan}^{- 1}{\text{1+tan}}^{- 1}\left(3\right) - {\text\tan}^{- 1}{\text{2+....+tan}}^{- 1}\left(24\right) - {\text\tan}^{- 1}\text{23}

\text{=cot}\left({\text\tan}^{- 1}24 - {\text\tan}^{- 1}1\right)
\text{=cot}\left({\text\tan}^{- 1}\dfrac{24 - 1}{1 + 24}\right)
$\text{=cot}\left({\text{cot}}^{- 1}\dfrac{25}{23}\right) = \dfrac{25}{23}$