What is tan⁻¹(a/b) – tan⁻¹((a–b)/(a+b)) equal to?

Using tan⁻¹x – tan⁻¹y = tan⁻¹((x–y)/(1+xy)): x = a/b, y = (a–b)/(a+b). x – y = a/b – (a–b)/(a+b) = [a(a+b) – b(a–b)]/[b(a+b)] = (a² + ab – ab + b²)/[b(a+b)] = (a²+b²)/[b(a+b)]. 1 + xy = 1 + (a/b)(a–b)/(a+b) = [b(a+b) + a(a–b)]/[b(a+b)] = (ab + b² + a² – ab)/[b(a+b)] = (a²+b²)/[b(a+b)]. So (x–y)/(1+xy) = 1, hence tan⁻¹(1) = π/4.