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

$\begin{array}{rcl} & & {\sin }^2{5}^0+{\sin }^2{10}^0 + {\sin }^2{15}^0 + \dots \dots +{\sin }^2{90}^0\\ & & \begin{array}{rcl}\mathrm{A}+\mathrm{B}& =& {90}^0 \Rightarrow {\sin }^2\mathrm{A}+{\sin }^2\mathrm{B}=1\end{array}\\ & =& {\sin }^2{5}^0+{\sin }^2{75}^0+{\sin }^2{10}^0+{\sin }^2{70}^0+.......\\ & & +.......+{\sin }^2{40}^0+{\sin }^2{50}^0+{\sin }^2{45}^0+{\sin }^2{90}^0\\ & =& 1+1+1+1+1+1+1+1+\frac{1}{2}+1\\ & =& 9\frac{1}{2}\end{array}$