${\cos }^2{5}^0 + {\cos }^2{10}^0 + {\cos }^{ 2}{15}^0+\dots \dots +{\cos }^{ 2}{90}^0 =$

$\begin{array}{rcl} & & {\cos }^2{5}^0 + {\cos }^2{10}^0 + {\cos }^{ 2}{15}^0+\dots \dots +{\cos }^{ 2}{90}^0\\ \mathrm{A}+\mathrm{B}& =& {90}^0 \Rightarrow {\cos }^2\mathrm{A}+{\cos }^2\mathrm{B}=1\\ & =& {\cos }^2{5}^0+{\cos }^2{85}^0+{\cos }^2{10}^0+{\cos }^2{80}^0+.......\\ & & ......+{\cos }^2{40}^0+{\cos }^2{50}^0+{\cos }^2{45}^0+{\cos }^2{90}^0\\ & =& 1+1+1+1+1+1+1+1+\frac{1}{2}\\ & =& 8\frac{1}{2}\end{array}$