`2"tan"(tan^(-1)(x)+tan^(-1)(x^3))`,where `x in R-{-1,1},` is equal to

Options

A. `(2x)/(1-x^2)`
B. `tan(2tan^(-1)x)`
C. `tan(cot^(-1)(-x)-cot^(-1)(x))`
D. `"tan"(2cot^(-1)x)`

Explanation

Let `tan^-1x = alpha => tanalpha = x`

Let `tan^-1x^3 = beta => tan beta = x^3`

Then,

`2tan(tan^-1x+tan^-1x^3) = 2tan(alpha+beta)`

`=2[(tanalpha+tanbeta)/(1-tanalphatanbeta)]`

`=2[(x+x^3)/(1-x(x^3))]`

`=2[(x(1+x^2))/(1-x^4)]`

`=2[(x(1+x^2))/((1-x^2)(1+x^2))]`

...