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

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Aicharya · Accepted Answer

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

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

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