If A is an acute angle, the simplified form of $$\frac{{\cos \left( {\pi - A} \right).\cot \left( {\frac{\pi }{2} + A} \right)\cos \left( { - A} \right)}}{{\tan \left( {\pi + A} \right)\tan \left( {\frac{{3\pi }}{2} + A} \right)\sin \left( {2\pi - A} \right)}}\,{\text{is:}}$$

$$\eqalign{ & \frac{{\cos \left( {\pi - A} \right).\cot \left( {\frac{\pi }{2} + A} \right)\cos \left( { - A} \right)}}{{\tan \left( {\pi + A} \right)\tan \left( {\frac{{3\pi }}{2} + A} \right)\sin \left( {2\pi - A} \right)}}\, \cr & = \frac{{\left( { - \cos A} \right) \times \left( { - \tan A} \right) \times \cos A}}{{\tan A \times \left( { - \cot A} \right) \times \left( { - \sin A} \right)}} \cr & = \frac{{{{\cos }^2}A}}{{\frac{{\cos A}}{{\sin A}} \times \sin A}} \cr & = \frac{{{{\cos }^2}A}}{{\cos A}} \cr & = \cos A \cr} $$