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

Explanation

Case I: `x gt 0`
2 tan^(-1) (cosec tan^(-1) x - tan cot^(-1) x)`
`= 2 tan^(-1) `(cosec(cosec^(-1) (sqrt(1 + x^(2)))/(x)) - tan (tan^(-1). (1)/(x)))`
`= 2 tan ^(-1) ((sqrt(1 + x^(2)))/(x) - (1)/(x))`
`= 2 tan^(-1) ((sec theta - 1)/(sin 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`
Case II: `x lt 0`
Let `y = -x`
`:. y gt 0`
`:. 2 tan^(-1) (cosec tan^(-1) x - tan cot^(-1) x)`
`= 2 tan^(-1) (cosec tan^(-1) (-y) - tan cot^(-1) (-y))`
`=2 tan^(-1) (cosec (-tan^(-1) y) - tan (pi - cot^(-1) y))`
`= 2 tan^(-1) (-cosec (tan^(-1) y) + tan (cot^(-1) y))`
`= 2 tan^(-1) (-cosec (cosec^(-1) (sqrt(1 + y^(2)))/(y)) + tan (tan^(-1).(1)/(y)))`
`= 2 tan^(-1) (-(sqrt1 + y^(2))/(y) + (1)/(y))`
`= 2 tan^(-1) ((sqrt(1 + y^(2))-1)/(y))`
`= -tan^(-1) y`
`= tan^(-1) (-y)`
`= tan^(-1) x`