The value of cot ( n = 1 23 cot - 1 ( 1 + k = 1 n 2 k ) ) is [2013]

Question

The value of $\cot (\sum_{n = 1}^{23} \cot^{- 1} (1 + \sum_{k = 1}^{n} 2 k))$ is [2013]

Smart Achievers · Accepted Answer

(2)

$\cot^{- 1} (1 + \sum_{k = 1}^{n} 2 k) = \cot^{- 1} [1 + n (n + 1)]$

$= \tan^{- 1} (\frac{(n + 1) - n}{1 + (n + 1) n}) = \tan^{- 1} (n + 1) - \tan^{- 1} n$

$∴$ $\sum_{n = 1}^{23} [\tan^{- 1} (n + 1) - \tan^{- 1} n] = \tan^{- 1} 24 - \tan^{- 1} 1 = \tan^{- 1} \frac{23}{25}$

$∴$ $\cot [\sum_{n = 1}^{23} \cot^{- 1} (1 + \sum_{k = 1}^{n} 2 k)] = \cot [\tan^{- 1} \frac{23}{25}] = \frac{25}{23}$

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