Question

Show that `2tan^-1 (-3) = (-pi)/2 + tan^-1 ((-4)/3)`

Answer 1

L.H.S. `2tan^-1 (-3) = -2tan^-1 (3)`

= `- cos^-1 [(1- (3)^2)/(1 + (3)^2)]`  ......`[because 2tan^-1x = cos^-1 ((1 - x^2)/(1 + x^2))]`

= `-cos^-1 ((1 - 9)/(1 + 9))`

= `- cos^-1 ((-8)/10)`

= `- cos^-1 ((-4)/5)`

= `- [pi - cos^-1 (4/5)]`

= `- pi + cos^-1  4/5`

= `- pi + tan^-1 (3/4)`  ......`[because cos^-1  4/5 = tan^-1  3/4]`

= `- pi + pi/2 - cot^-1 (3/4)`  ......`[tan^-1x = pi/2 - cot^-1x]`

= `(-pi)/2 - cot^-1 (3/4)`

= `(-pi)/2 - tan^-1 (4/3)`  .......`[because tan^-1x = cot^-1  1/x]`

= `(-pi)/2 + tan^-1 (- 4/3)`  R.H.S

Hence proved.