The value ofcot⁡∑$$n=123$$ $$cot$$−$$1$$⁡$$1+$$∑$$k=1n$$ $$2k$$, is

Question

The value ofcot⁡∑$$n=123$$ $$cot$$−$$1$$⁡$$1+$$∑$$k=1n$$ $$2k$$, is

Infinity Learn · Accepted Answer

Clearly,$$cot$$−$$1$$⁡$$1+$$∑$$k=1n$$ $$2k=$$$$cot$$−$$1$$⁡$$1+2$$∑$$k=1n$$ $$k=$$$$cot$$−$$1$$⁡(1+n(n+1)) $$=$$$$tan$$−$$1$$⁡$$11+n(n+1)=$$$$tan$$−$$1$$⁡(n+1)−$$n1+n(n+1)$$ $$=$$$$tan$$−$$1$$⁡(n+1)−$$tan$$−$$1$$⁡n∴ ∑$$n=123$$ $$cot$$−$$1$$⁡$$1+$$∑$$k=1n$$ $$2k$$ $$=$$∑$$n=123$$ $$tan$$−$$1$$⁡(n+1)−$$tan$$−$$1$$⁡n $$=$$$$tan$$−$$1$$⁡$$24$$−$$tan$$−$$1$$⁡$$1$$ $$=$$$$tan$$−$$1$$⁡$$24$$−$$11+24$$×$$1=$$$$tan$$−$$1$$⁡$$2325=$$$$cot$$−$$1$$⁡$$2523Hence$$, $$cot$$ ∑$$n=123$$ $$cot$$−$$1$$⁡$$1+$$∑$$k=1n$$ $$2k=$$$$cot$$⁡$$cot$$−$$1$$⁡$$2523=2523$$

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

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The value ofcot⁡∑n=123 cot−1⁡1+∑k=1n 2k, is

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