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

\(\begin{aligned} & \cot \left(\sum_{n=1}^{23} \cot ^{-1}\left(1+\sum_{k=1}^n 2 k\right)\right) \\ & =\cot \left(\sum_{n=1}^{23} \cot ^{-1}\left(1+2 \times \frac{n(n+1)}{2}\right)\right) \\ & =\cot \left(\sum_{n=1}^{23} \cot ^{-1}(1+n(n+1))\right)\end{aligned}\)
\(\begin{aligned} & =\cot \left(\sum_{n=1}^{23} \tan ^{-1}\left(\frac{1}{1+\mathrm{n}(\mathrm{n}+1)}\right)\right) \\ & =\cot \left(\sum_{\mathrm{n}=1}^{23} \tan ^{-1}\left(\frac{\mathrm{n}+1-\mathrm{n}}{1+\mathrm{n}(\mathrm{n}+1)}\right)\right) \\ & =\cot \left(\sum_{\mathrm{n}=1}^{23} \tan ^{-1}(\mathrm{n}+1)-\sum_{\mathrm{n}-1}^{23} \tan ^{-1} \mathrm{n}\right) \\ & =\cot \left[\left(\tan ^{-1}(2)+\tan ^{-1}(3)+\ldots+\tan ^{-1}(24)\right)\right. \\ & =\cot \left(\tan ^{-1}(24)-\tan ^{-1}(1)\right) \\ & =\cot \left(\tan ^{-1}\left(\frac{24-1}{1+24(1)}\right)\right) \\ & =\cot \left(\tan ^{-1}\left(\frac{23}{25}\right)\right) \\ & =\cot \left(\cot ^{-1}\left(\frac{25}{23}\right)\right) \\ & =\frac{25}{23}\end{aligned}\)