Prove that `2 tan^(-1) (cosec tan^(-1) x - tan cot^(-1) x) = tan^(-1) x (x != 0)`

Options

A. `cot^(-1) x`
B. `cot^(-1).(1)/(x)`
C. `tan^(-1) x`
D. none of these

Answer

C

Explanation

`2 tan^(-1 (cosec tan^(-1) x - tan cot^(-1) x)`
`= 2 tan^(-1) [cosec {cosec^(-1) (sqrt(1 + x^(2)))/(x)} - tan^(-1) {tan^(-1) ((1)/(x))}]`
`= 2 tan^(-1) [sqrt((1 + x^(2))/(x)) - (1)/(x)] = 2 tan^(-1) [(sqrt(1 + x^(2)) - 1)/(x)]`
`= 2 tan^(-1) [(sec theta -1)/(tan theta)]` [Putting `x = tan theta`]
`= 2 tan^(-1) [(1 - cos theta)/(sin theta)] = 2 tan^(-1) [(2 sin^(2).(theta)/(2))/(2 sin.(theta)/(2) cos.(theta)/(2))]`
`= 2 tan^(-1) tan.(theta)/(2) = 2 xx (theta)/(2) = theta = tan^(-1) x`