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`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)`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)` |
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Answer» Correct Answer - A::B::C 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))` |
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