Tính tổng \(S = {\sin ^2}{5^0} + {\sin ^2}{10^0} + {\sin ^2}{15^0} + ... + {\sin ^2}{85^0}\)

\(\begin{array}{l}S = {\sin ^2}{5^0} + {\sin ^2}{10^0} + {\sin ^2}{15^0} + ... + {\sin ^2}{85^0}\\ = \left( {{{\sin }^2}{5^0} + {{\sin }^2}{{85}^0}} \right) + \left( {{{\sin }^2}{{10}^0} + {{\sin }^2}{{80}^0}} \right) + ... + \left( {{{\sin }^2}{{40}^0} + {{\sin }^2}{{50}^0}} \right) + {\sin ^2}{45^0}\\ = \left[ {{{\cos }^2}\left( {{{180}^0} - {5^0}} \right) + {{\sin }^2}{{85}^0}} \right] + \left[ {{{\cos }^2}\left( {{{180}^0} - {{10}^0}} \right) + {{\sin }^2}{{80}^0}} \right] + ... + \left[ {{{\cos }^2}\left( {{{180}^0} - {{40}^0}} \right) + {{\sin }^2}{{50}^0}} \right] + {\sin ^2}{45^0}\\ = \left( {{{\cos }^2}{{85}^0} + {{\sin }^2}{{85}^0}} \right) + \left( {{{\cos }^2}{{80}^0} + {{\sin }^2}{{80}^0}} \right) + ... + \left( {{{\cos }^2}{{50}^0} + {{\sin }^2}{{50}^0}} \right) + {\sin ^2}{45^0}\\ = 1 + 1 + .. + 1 + {\left( {\frac{{\sqrt 2 }}{2}} \right)^2} = 8 + {\left( {\frac{{\sqrt 2 }}{2}} \right)^2}\\ = \frac{{17}}{2}\end{array}\)