Question

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

Answer 1

`sum_(k=1)^n 2k=2sum_(k=1)^n k`
`=2*(n(n+1)/2)`
`=n(n+1)`
`cot^(-1)[1+n)n+1)]=tan^(-1)[1/(1+n(n+1))]`
`tan_(-1)[(n+1-n)/(1+(n+1)n)]`
`tan_(-1) ((x-y)/(1+xy))`
`=tan^(-1)(n+1)-tan^(-1)n`
`cot^(-1){sum_(n=1)^(23)(tan^(-1)(n+1)-tan^(-1)n)}`
`cot^(-1){tan_(-1)(n+1)-tan^(-1)n`
`cot^(-1){tan_(-1)23/25}`
`cot^(-1)[cos^(-1)25/23]=25/23`
option b is correct.