$\begin{array}{l} {\cos ^2}\left( {\dfrac{\pi }{4} - x} \right) = {\cos ^2}\left( {\dfrac{\pi }{2} - \left( {\dfrac{\pi }{4} + x} \right)} \right) = {\sin ^2}\left( {\dfrac{\pi }{4} + x} \right)\\ \to A = \dfrac{{1 - 2{{\sin }^2}x}}{{2\cot \left( {\dfrac{\pi }{4} + x} \right){{\cos }^2}\left( {\dfrac{\pi }{4} - x} \right)}} = \dfrac{{\cos 2x}}{{2\cot \left( {\dfrac{\pi }{4} + x} \right){{\sin }^2}\left( {\dfrac{\pi }{4} + x} \right)}}\\ \to A = \dfrac{{\cos 2x}}{{2\cos \left( {\dfrac{\pi }{4} + x} \right)\sin \left( {\dfrac{\pi }{4} + x} \right)}} = \dfrac{{\cos 2x}}{{\sin \left( {\dfrac{\pi }{2} + 2x} \right)}} = \dfrac{{\cos 2x}}{{\cos 2x}} = 1 \end{array}$
