Find the sum `cos e c^(-1)sqrt(10)+cos e c^(-1)sqrt(50)+cos e c^(-1)sqrt(170)++cos e c^(-1)sqrt((n^2+1)(n^2+2n+2))`

Explanation

Let `S = cosec^-1sqrt10+cosec^-1sqrt50+cosec^-1sqrt170+...+cosec^-1sqrt((n^2+1)(n^2+2n+2))`

Here, `T_n = cosec^-1sqrt((n^2+1)(n^2+2n+2))`

Let `cosec^-1sqrt((n^2+1)(n^2+2n+2)) = theta`

`=>cosec theta = sqrt((n^2+1)(n^2+2n+2))`

`=>cosec^2theta = (n^2+1)(n^2+2n+2)`

`=>cosec^2theta = (n^2+1)(n^2+1+2n+1)`

`=>1+cot^2theta = (n^2+1)(n^2+1)+2n(n^2+1)+n^2+1`

`=>1+cot^2theta = (n^2+1+n)^2+1`

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