$\frac{{\cot }^2{15}^0-1}{{\cot }^{2}15+1}=$

$\frac{{\cot }^2{15}^{\circ }-1}{{\cot }^2{15}^{\circ }+1}=\frac{\frac{{\cos }^2{15}^{\circ }}{{\sin }^2{15}^{\circ }}-1}{\frac{{\cos }^2{15}^0}{{\sin }^2{15}^{\circ }}+1}=\frac{{\cos }^2{15}^{\circ }-{\sin }^2{15}^{\circ }}{{\cos }^2{15}^{\circ }+{\sin }^2{15}^{\circ }}$ $=\frac{\cos 2\left({15}^{\circ }\right)}{1}=\cos {30}^{\circ }=\frac{\sqrt{3}}{2}$