$\cot \left(\mathrm{A}+{15}^{\circ }\right)-\tan \left(\mathrm{A}-{15}^{\circ }\right)=$

$\begin{array}{l}=\frac{\cos \left(\mathrm{A}+{15}^{\circ }\right)}{\sin \left(\mathrm{A}+{15}^{\circ }\right)}-\frac{\sin \left(\mathrm{A}-{15}^{\circ }\right)}{\cos \left(\mathrm{A}-{15}^{\circ }\right)}=\frac{\cos \left(\mathrm{A}+{15}^{\circ }\right)\cos \left(\mathrm{A}-{15}^{\circ }\right)-\sin \left(\mathrm{A}+{15}^{\circ }\right)\sin \left(\mathrm{A}-{15}^{\circ }\right)}{\sin \left(\mathrm{A}+{15}^{\circ }\right)\cos \left(\mathrm{A}-{15}^{\circ }\right)}\\ =\frac{\cos 2\mathrm{A}}{\frac{1}{2}\left(\sin 2\mathrm{A}+\sin {30}^{\circ }\right)}=\frac{4\cos 2\mathrm{A}}{1+2\sin 2\mathrm{A}}\end{array}$