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Question

Given that 0 < a < b < c, evaluate the sum: cot⁻¹((ab + b)/(a − b)) + cot⁻¹((bc + 1)/(b − c)) + cot⁻¹((ca + 1)/(c − a)).

Accepted Answer

2π. Reading the terms as cot^-1((ab+1)/(a-b)) + cot^-1((bc+1)/(b-c)) + cot^-1((ca+1)/(c-a)) with 0<a<b<c, the first two denominators are negative. Using cot^-1((xy+1)/(y-x)) = cot^-1 x - cot^-1 y and the principal range (0,pi), the negative-argument terms each contribute an extra pi, so the telescoping sum totals 2*pi.

Given that 0 < a < b < c, evaluate the sum: cot⁻¹((ab + b)/(a − b)) + cot⁻¹((bc + 1)/(b − c)) + cot⁻¹((ca + 1)/(c − a)).

Solution

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