Prove that: `tan^-1((c_1x-y)/(c_1y+x))+tan^-1((c_2-c_1)/(1+c_2c_1))+tan^-1((c_3-c_2)/(1+c_3c_2))` +`.......+tan^-1((c_n-c_(n-1))/(1+c_nc_(n-1)))+tan^-1(1/c_n)=tan^-1(x/y)`

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`