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

Engineering

Mathematics

Properties of Inverse Trigonometric Function

Question

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

$\frac{25}{23}$

$\frac{23}{24}$

$\frac{23}{25}$

$\frac{24}{23}$

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$\sum_{k = 1}^{n} 2 k = n (n + 1)$

$∴ \cot^{- 1} [1 + n (n + 1)] = \tan^{- 1} (\frac{1}{1 + n (n + 1)}) = \tan^{- 1} (\frac{(n + 1) - n}{1 + n (n + 1)}) = \tan^{- 1} (n + 1) - \tan^{- 1} n$

$∴ \sum_{n = 1}^{23} \cot^{- 1} (1 + \sum_{k = 1}^{n} 2 k) = \cot [\sum_{n = 1}^{23} \tan^{- 1} (n + 1) - \tan^{- 1} n] = \cot [\tan^{- 1} 24 - \tan^{- 1} 1]$

$= \cot (\tan^{- 1} \frac{23}{25}) = \frac{25}{23}$