The value of `cot[{Sigma_(n=1)^(23) {cot^(-1))1+Sigma_(k=1)^(n) 2k}]` isA. `23/25`B. `25/23`C. `23/24`D. `24/23`

The value of `cot[{Sigma_(n=1)^(23) {cot^(-1))1+Sigma_(k=1)^(n) 2k}]`

1.	The value of `cot[{Sigma_(n=1)^(23) {cot^(-1))1+Sigma_(k=1)^(n) 2k}]` isA. `23/25`B. `25/23`C. `23/24`D. `24/23`
Answer» 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`

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