${\left(\frac{\cos A+\cos B}{\sin A-\sin B}\right)}^{2005}+{\left(\frac{\sin A+\sin B}{\cos A-\cos B}\right)}^{2005}=$

${\left[\frac{2\cos \left(\frac{A+B}{2}\right)\cdot \cos \left(\frac{A-B}{2}\right)}{2\cos \left(\frac{A+B}{2}\right)\cdot \sin \left(\frac{A-B}{2}\right)}\right]}^{2005}+{\left[\frac{2\sin \left(\frac{A+B}{2}\right)\cdot \cos \left(\frac{A-B}{2}\right)}{-2\sin \left(\frac{A+B}{2}\right)\cdot \sin \left(\frac{A-B}{2}\right)}\right]}^{2005}$

${\left[\cot \left(\frac{A-B}{2}\right)\right]}^{2005}+{\left[-\cot \left(\frac{A-B}{2}\right)\right]}^{2005}$

${\cot }^{2005}\left(\frac{A-B}{2}\right)-{\cot }^{2005}\left(\frac{A-B}{2}\right)$

$=0$