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`

Options

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

Explanation

Clearly
`cot^(-1)(1+underset(k=1)overset(n)Sigma 2k)=cot^(-1)1+2underset(k=1)overset(n_Sigmak)=cot^(-1)1+n(n+1)`
`=tan^(-1)(1)/(1+n(n+1)}=(tan^(-1)(n+1)-n)/(1+n(n+1))`
`tan^(-1)(n1)-tan^(-1)n`
`therefore underset(n=1)overset(23)Sigma(tan^(-1)(n+1)-tan^(-1)n)`
`tan^(-1)24-tan^(-1)1`
`tan^(-1)(24-1)/(1+24xx1)=tan^(-1)(23)/(25)=cot^(-1)25/23`
Hence `cot[underset(n=1)overset(23)Sigma{cot^(-1)(1+underset(k=1)overset(n)Sigma 2k)}]=cot^(-1)25/23=25/23`