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

Solution :
$$\eqalign{ & {\cot ^{ - 1}}\left( {1 + \sum\limits_{k = 1}^n {2k} } \right) \cr & = {\cot ^{ - 1}}\left[ {1 + n\left( {n + 1} \right)} \right] \cr & = {\tan ^{ - 1}}\left[ {\frac{{\left( {n + 1} \right) - n}}{{1 + \left( {n + 1} \right)n}}} \right] \cr & = {\tan ^{ - 1}}\left( {n + 1} \right) - {\tan ^{ - 1}}n \cr & \therefore \,\,\sum\limits_{n = 1}^{23} {\left[ {{{\tan }^{ - 1}}\left( {n + 1} \right) - {{\tan }^{ - 1}}n} \right]} \cr & = {\tan ^{ - 1}}24 - {\tan ^{ - 1}}1 \cr & = {\tan ^{ - 1}}\frac{{23}}{{25}} \cr & \therefore \,\,\cot \left[ {\sum\limits_{n = 1}^{23} {{{\cot }^{ - 1}}\left( {1 + \sum\limits_{k = 1}^n {2k} } \right)} } \right] \cr & = \cot \left[ {{{\tan }^{ - 1}}\frac{{23}}{{25}}} \right] \cr & = \frac{{25}}{{23}} \cr} $$