The value of  cot⁡∑n=119 cot−1⁡1+∑p=1n 2p is

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