${\cos }^2\left(\theta -{45}^0\right)+{\cos }^2\left(\theta +{15}^0\right)-{\cos }^2\left(\theta -{15}^0\right) =$

$\begin{array}{rcl} & & {\cos }^2\left(\mathrm{\theta }-{45}^0\right)+{\cos }^2\left(\mathrm{\theta }+{15}^0\right)-{\cos }^2\left(\mathrm{\theta }-{15}^0\right)\\ & & {\cos }^2({45}^0-\mathrm{\theta })+\sin (\mathrm{\theta }+{15}^0+\mathrm{\theta }-{15}^0)\sin (\mathrm{\theta }-{15}^0-\mathrm{\theta }-{15}^0)\\ & =& \frac{1+\cos 2({45}^0-\mathrm{\theta })}{2}+\sin 2\mathrm{\theta } \sin (-{30}^0)\\ & =& \frac{1}{2}+\frac{1}{2}\cos ({90}^0-2\mathrm{\theta })-\frac{1}{2}\sin 2\mathrm{\theta }\\ & =& \frac{1}{2}+\frac{1}{2}\sin 2\mathrm{\theta }-\frac{1}{2}\sin 2\mathrm{\theta }\\ & =& \frac{1}{2}\end{array}$