If `alpha in (-(pi)/(2), 0)`, then find the value of `tan^(-1) (cot alpha) - cot^(-1) (tan alpha)`

Answer

`-pi`

Explanation

`tan^(-1) (cot alpha) - cot^(-1) (tan alpha)`
`= tan^(-1) ((1)/(tan alpha)) - ((pi)/(2) - tan^(-1) (tan alpha))`
`= -(pi)/(2) + (tan^(-1) ((1)/(tan alpha)) + tan^(-1) (tan alpha))`
`= -(pi)/(2) - (pi)/(2) " " ("as " (-pi)/(2) lt alpha lt 0)`