If `x_1=2tan^(-1)((1+x)/(1-x)),x_2=sin^(-1)((1-x^2)/(1+x^2))` , where `x in (0,1),` then `x_1+x_2` is equal to 0 (b) `2pi` (c) `pi` (d) none of these

Options

A. 0
B. `2pi`
C. `pi`
D. none of these

Answer

C

Explanation

`x_(1) = 2 tan^(-1) ((1 + x)/(1 - x))`
and `x_(2) = sin^(-1) ((1 - x^(2))/(1 + x^(2)))`
`= tan^(-1) ((1 - x^(2))/(2x))`
Now `(1 + x)/(1 -x) gt 1`
`rArr x_(1) = pi + tan^(-1) ((2((1 + x)/(1 -x)))/(1 -((1+ x)/(1 -x))^(2)))`
`= pi + tan^(-1) ((1 -x^(2))/(-2x))`
`= pi - tan^(-1) ((1 -x^(2))/(2x))`
`rArr x_(1) + x_(2) = pi`