Prove that `tan^(-1) x + tan^(-1).(1)/(x) = {(pi//2,"if" x gt 0),(-pi//2," if " x lt 0):}`

Explanation

We know that `tan^(-1) ((1)/(x)) = {(cot^(-1) x,x gt 0),(-pi + cot^(-1) x,x lt 0):}`
`rArr tan^(-1) x + "tan"^(-1)(1)/(x) = {(tan^(-1) x + cot^(-1) x,x gt 0),(-pi + cot^(-1) x + tan^(-1) x,x lt 0):}`
`={((pi)/(2),x gt0),(-pi + (pi)/(2),x lt 0):}`
`= {((pi)/(2),x gt 0),((-pi)/(2),x lt 0):}`