Calculation of a Minimum Distance

$\large \lim_{\theta \to \pi/4} {\frac{{\mathop {(\cos \theta )}\nolimits^{\frac{{\sin \theta }}{{\cos \theta - \sin \theta }}} }}{{\mathop {(\sin \theta )}\nolimits^{\frac{{\cos \theta }}{{\cos \theta - \sin \theta }}} }}} = a \sqrt b$

If the equation above holds true for square-free positive integer $b$ , find $\lfloor a \rfloor +b$ .

Notations : $\lfloor \cdot \rfloor$ denotes the floor function.