The value of `cot(sum_(n=1)^(23)cot^-1(1+sum_(k=1)^n2k))` is

Options

A. `(23)/(25)`
B. `(25)/(23)`
C. `(23)/(24)`
D. `(24)/(23)`

Answer

B

Explanation

`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)`