`sum_(r=1)^nsin^(-1)((sqrt(r)-sqrt(r-1))/(sqrt(r(r+1))))i se q u a lto` `tan^(-1)(sqrt(n))-pi/4` `tan^(-1)(sqrt(n+1))-pi/4` `tan^(-1)(sqrt(n))` (d) `tan^(-1)(sqrt(n)+1)`

Options

A. `tan^(-1) (sqrtn) - (pi)/(4)`
B. `tan^(-1) (sqrt(n+1)) - (pi)/(4)`
C. `tan^(-1)(sqrtn)`
D. `tan^(-1) (sqrt(n+1))`

Answer

C

Explanation

`sin^(-1) ((sqrtr - sqrt(r -1))/(sqrt(r( r+1)))) = tan^(-1) ((sqrtr - sqrt(r -1))/(1 + sqrt(r (r -1))))`
`:. underset(r =1)overset(n) sum sin^(-1) ((sqrtr - sqrt(r -1))/(sqrt(r (r + 1))))`
`= underset(r =1)overset(n) sum (tan^(-1) sqrtr - tan^(-1) sqrt(r -1)) = tan^(-1) sqrtn`