`2"tan"(tan^(-1)(x)+tan^(-1)(x^3)),w h e r ex in R-{-1,1},` is equal to `(2x)/(1-x^2)` `t(2tan^(-1)x)` `tan(cot^(-1)(-x)-cot^(-1)(x))` `"tan"(2cot^(-1)x)`

Options

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

Answer

A, B, C

Explanation

Let `tan^(-1) x = alpha and tan^(-1) x^(3) = beta`
`tan alpha = x and tan beta = x^(3)`
`:. 2 tan (alpha + beta) = (2(tan alpha + tan beta))/(1-tan alpha tan beta) = 2[(x + x^(3))/(1 -x^(4))] = (2x)/(1 -x^(2))`
Also, `(2x)/(1 -x^(2)) = (2 tan alpha)/(1 -tan^(2) alpha) = tan 2 alpha = tan (2 tan^(-1) x)`
`= tan (2 ((pi)/(2) - cot^(-1) x))`
`= tan (pi -cot^(-1) x - cot^(-1) x)`
`= tan (cot^(-1) (-x) - cot^(-1) (x))`