The key identity: tan^(-1)((a-b)/(1+ab)) = tan^(-1)(a) - tan^(-1)(b) when ab > -1. To telescope a sum, express each term as a difference of consecutive inverse tangents. Common pattern: tan^(-1)(1/(1+n(n+1))) = tan^(-1)(n+1) - tan^(-1)(n). Sum from n=1 to N = tan^(-1)(N+1) - pi/4.