Question

Show that

(b - a)/(1 - b²) < tan–1 b – tan^–1 a < (b - a)/(1 + a²), if 0 < a < b and deduce that  π/4 + 3/25 < tan^-1 4/3 < π/4 + 1/6.

Answer 1

Let f (x) = tan^–1x is continuous in [a, b], a > 0.

Hence f ′ (x) = 1/(1 + x²),  f(x) is differentiable in (a, b).

Applying Lagrange’s theorem, we get