`tan^(-1)((1)/(4))+tan^(-1)((2)/(11))=`

Explanation

Here,
`tan^(_1) ((a_(1) x- y)/(a + a_(1) y)) = tan^(-1) ((a_(1) - (y)/(x))/(1 + a_(1) (y)/(x))) = tan^(-1) a_(1) - tan^(-1) (y)/(x)`
`tan^(-1) ((a_(2) -a_(1))/(1 + a_(2) a_(1))) = tan^(-1) a_(2) - tan^(-1) a_(1)`
`{:(tan^(-1) ((a_(3) - a_(2))/(1 + a_(3) a_(2))) = tan^(-1) a_(3) - tan^(-1) a_(2)),(vdots),(tan^(-1) ((a_(n) - a_(n -1))/(1 + a_(n) a_(n -1))) = tan^(-1) a_(n) - tan^(-1) a_(n -1)):}`
`tan^(-1) ((1)/(a_(n))) = cot^(-1) a_(n)`
Adding, we get
`L.H.S. = tan^(-1) a_(n) - tan^(-1) (y)/(x) = (pi)/(2) - tan^(-1) (y)/(x)`
`= cot^(-1) (y)/(x) = tan^(-1) (x)/(y) = R.H.S`