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The value of $\cot [\sum_{n = 1}^{19} \cot^{- 1} \{1 + \sum_{p = 1}^{n} 2 p\}]$ is

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a

2223

b

2322

c

2119

d

1921

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detailed solution

Correct option is C

1+∑p=1n 2p=1+n(n+1)=n2+n+1∴ cot−1⁡1+∑p=1n 2p=cot−1⁡n2+n+1=tan−1⁡11+n(n+1)=tan−1⁡(n+1)−n1+n(n+1)=tan−1⁡(n+1)−tan1⁡n⇒∑n=119 cot−1⁡1+∑p=1n 2p=∑n=119 tan−1⁡(n+1)−tan−1⁡n=tan−1⁡20−tan−1⁡1=tan−1⁡1921⇒cot⁡∑n=119 cot−1⁡1+∑p=1n 2p=cot⁡tan-11921=cot⁡cot−1⁡2119=2119

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