Let `tan^(-1)y=tan^(-1)x+tan^(-1)((2x)/(1-x^2))` , where `|x|<1/(sqrt(3))` . Then a value of y is : (1) `(3x-x^3)/(1-3x^2)` (2) `(3x+x^3)/(1-3x^2)` (3) `(3x-x^3)/(1+3x^2)` (4) `(3x+x^3)/(1+3x^2)`

Options

A. `(3x -x^(3))/(1 -3x^(2))`
B. `(3x + x^(3))/(1 - 3x^(2))`
C. `(3x -x^(3))/(1 + 3x^(2))`
D. `(3x + x^(3))/(1 + 3x^(2))`

Answer

B

Explanation

`(-1)/(sqrt3) lt x lt (1)/(sqrt3)`
Let `x = tan theta`
`:. (-pi)/(6) lt theta lt (pi)/(6)`
`:. Tan^(-1) y = theta + tan^(-1) tan 2 theta = theta + 2theta = 3 theta`
`:. Y = tan 3 theta = (3 tan theta - tan^(3) theta)/(1-3 tan^(2) theta)`
`:. y = (3x -x^(3))/(1 -3x^(2))`