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The value of cot(sum(n=1)^23 cot^-1(1+su...

The value of `cot(sum_(n=1)^23 cot^-1(1+sum_(k=1)^n 2k))` is (a) `23/25` (b) `25/23` (c) `23/24` (d) `25/26`

A

`(23)/(25)`

B

`(25)/(23)`

C

`(23)/(24)`

D

`(24)/(23)`

Text Solution

Verified by Experts

The correct Answer is:
B
`cot(underset(n =1)overset(23)sum cot^(-1) (n^(2) + n + 1))`
`= cot(underset(n =1)overset(23)sum tan^(-1) ((n + 1 -n)/(1+ n (n + 1))))`
`= cot (underset(n=1)overset(23)sum (tan^(-1) (n +1) - tan^(-1)n))`
`=cot (tan^(-1) 24 -tan^(-1) 1) rArr cot (tan^(-1) ((23)/(25)))`
`= (25)/(23)`
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