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The value of cotsum(n=1)^(19)cot^(-1)(1+...

The value of `cotsum_(n=1)^(19)cot^(-1)(1+sum_(p=1)^(n)2p)` is equal to (a) `(21)/(19)` (b) `(19)/(21)` (c) `-(19)/(21)` (d) `-(21)/(19)`

A

`(23)/(22)`

B

`(19)/(21)`

C

`(22)/(23)`

D

`(21)/(19)`

Text Solution

Verified by Experts

The correct Answer is:
D
`cot (sum_(n=1)^(19)cot^(-1)(1+sum_(p=1)^(n)2p))`
`sum_(P=1)^(n)2P=2+4+6+.......+2n=n^(2)+n`
`sum_(n=1)^(19)cot^(-1)(n^(2)+n+1)`
`sum_(n=1)^(19)tan^(-1)(((n+1)-(n))/(1+n(n+1)))`
`sum_(n=1)^(19)tan^(-1)(n+1)-tan^(-1)n=tan^(-1)20-(pi)/(4)`
`therefore cot(tan^(-1)20-(pi)/(4))=(1)/(tan(tan^(-1)20-(pi)/(4)))=(1+20)/(20-1)=(21)/(19)`
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