Question

`4tan^-1  1/5-tan^-1  1/239=pi/4`

Answer 1

LHS = `4tan^-1  1/5-tan^-1  1/239`

`=2tan^-1{(2xx1/5)/(1-(1/5)^2)}-tan^-1  1/239`     `[because2tan^-1x=tan^-1{(2x)/(1-x^2)}]`

`=2tan^-1{(2/5)/(24/25)}-tan^-1  1/239`

`=2tan^-1  5/12-tan^-1  1/239`

`=tan^-1{(2xx5/12)/(1-(5/12)^2)}-tan^-1  1/239`    `[because2tan^-1x=tan^-1{(2x)/(1-x^2)}]`

`=tan^-1{(5/6)/(119/144)}-tan^-1  1/239`

`=tan^-1  120/119-tan^-1  1/239`

`=tan^-1((120/119-17/239)/(1+120/119xx1/239))`      `[becausetan^-1x-tan^-1y=tan^-1((x-y)/(1+xy))]`

`=tan^-1 1=pi/4=`RHS