$$\cot \left( {\sum\limits_{n = 1}^{23} {{{\cot }^{ - 1}}\left( {1 + \sum\limits_{k = 1}^n {2k} } \right)} } \right)$$

$$ = \cot \left( {\sum\limits_{n = 1}^{23} {{{\cot }^{ - 1}}\left( {1 + 2 \times {{n(n + 1)} \over 2}} \right)} } \right)$$

$$ = \cot \left( {\sum\limits_{n = 1}^{23} {{{\cot }^{ - 1}}(1 + n(n + 1))} } \right)$$

$$ = \cot \left( {\sum\limits_{n = 1}^{23} {{{\cot }^{ - 1}}\left( {{{n(n + 1) + 1} \over {(n + 1) - n}}} \right)} } \right)$$

$$ = \cot \left( {\sum\limits_{n = 1}^{23} {{{\cot }^{ - 1}}n - {{\cot }^{ - 1}}(n + 1)} } \right)$$

$$ = \cot (({\cot ^{ - 1}}1 + {\cot ^{ - 1}}2 + {\cot ^{ - 1}}3 + .... + {\cot ^{ - 1}}23) - ({\cot ^{ - 1}}2 + {\cot ^{ - 1}}3 + .... + {\cot ^{ - 1}}23 + {\cot ^{ - 1}}24))$$

$$ = \cot ({\cot ^{ - 1}}1 - {\cot ^{ - 1}}24)$$

$$ = \cot \left( {{{\cot }^{ - 1}}{{24 \times 1 + 1} \over {24 - 1}}} \right) = {{25} \over {23}}$$