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Let f(x)=sin^(-1)((2x)/(1+x^(2))) and g(...

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

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))`

Text Solution

Verified by Experts

The correct Answer is:
C
`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))`
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