A geometry problem by cool dude

Geometry Level 4

\[ \large (\tan A - \cos B )^2 + ( \cot A - \sin B)^2 \]

Let \(A\) and \(B\) be real numbers. If the minimum value of the expression can be expressed as \( b - a \sqrt a\), where \(a\) and \(b\) are positive integers, find the value of \(a+b\).