An Acute Cotangent Window

In an acute triangle $ABC$, the sides opposite $A, B, C$ are $a, b, c$, and

1. Prove that $\cos B = 2\cos^2 A - 1$. 2. If $m < \dfrac{1}{\tan A} - \dfrac{1}{\tan B} < n$ holds for all such triangles, find the minimum value of $n - m$.