Let `f(x)=sin^(-1)((2x)/(1+x^(2)))` and `g(x)=cos^(-1)((x^(2)-1)/(x^(2)+1))`. Then tha value of f(10)-g(100) is equal to

Options

A. `pi-2(tan^(-1)(10)+tan^(-1)(100))`
B. 0
C. `2(tan^(-1)(100)-tan^(-1)(10))`
D. `2(tan^(-1)(10)-tan^(-1)(100))`

Answer

C

Explanation

`f(x)=sin^(-1)((2x)/(1+x^(2)))=pi-2 tan^(-1)x`, for `x ge 1`
and `g(x)=cos^(-1)((x^(2)-1)/(x^(2)+1))`
`=pi-cos^(-1)((1-x^(2))/(1+x^(2)))`
`=pi-cos^(-1)((1-x^())/(1+x^(2)))`
`=pi -2 tan^(-1)x`, for `x ge 0`
Now f(10)-g(100)
`=(pi-2tan^(-1)(10))-(pi-2tan^(-1)(100))`
`=2(tan^(-1)(100)-tan^(-1)(10))`