If `x<0,t h e ntan^(-1)x` is equal to `-pi+cot^(-1)1/x` (b) `sin^(-1)x/(sqrt(1+x^2))` `-cos^(-1)1/(sqrt(1+x^2))` (d) `-cos e c^(-1)(sqrt(1+x^2))/x`

Options

A. `-pi + cot^(-1).(1)/(x)`
B. `sin^(-1).(x)/(sqrt(1 + x^(2)))`
C. `-cos^(-1).(1)/(sqrt(1 + x^(2)))`
D. `- cosec^(-1).(sqrt(1 + x^(2)))/(x)`

Answer

A, B, C

Explanation

`tanx, x lt 0`
Let `x = -y, y gt 0`
`:. Tan^(-1) x = - tan^(-1) y`
`= -cos^(-1).(1)/(sqrt(1 + y^(2)))` ...(i)
`= -sin^(-1).(y)/(sqrt(1 + y^(2)))`..(ii)
`= -cosec^(-1).(sqrt(1 + y^(2)))/(y)`...(iii)
`=-cot^(-1).(1)/(y)` ...(iv)
From (i) `tan^(-1) x = - cos^(-1).(1)/(sqrt(1 + x^(2)))`
From (ii), `tan^(-1) x = -sin^(-1).(-x)/(sqrt(1 + x^(2))) = sin^(-1).(x)/(sqrt(1 + x^(2)))`
From (iii), `tan^(-1) x = - cosec^(-1).(sqrt(1 + x^(2)))/(-x) = cosec^(-1).(sqrt(1 + x^(2)))/(x)`
From (iv), `tan^(-1) x = - cot^(-1).(1)/(y)`
`= -cot^(-1).(1)/(-x) = -(pi - cot^(-1).(1)/(x)) = -pi + cot^(-1).(1)/(x)`