Tính nhanh gt biểu thức cos^2 15 +cos^2 15 +cos^2 35+cos^45+Cos^55+cos^65+cos^75 sin^2 10 – sin^2 20+sin^2 30-sin^2 40 -sin^2 50 -sin^2 70 +sin^2 80

Đáp án:

$\begin{array}{l}
A = {\cos ^2}{15^0} + {\cos ^2}{25^0} + {\cos ^2}{35^0} + {\cos ^2}{45^0}\\
 + {\cos ^2}{55^0} + {\cos ^2}{65^0} + {\cos ^2}{75^0}\\
 = {\sin ^2}\left( {{{90}^0} – {{15}^0}} \right) + {\sin ^2}\left( {{{90}^0} – {{25}^0}} \right) + {\sin ^2}\left( {{{90}^0} – {{35}^0}} \right)\\
 + {\cos ^2}{45^0} + {\cos ^2}{55^0} + {\cos ^2}{65^0} + {\cos ^2}{75^0}\\
 = {\sin ^2}{75^0} + {\sin ^2}{65^0} + {\sin ^2}{55^0}\\
 + \dfrac{1}{2} + {\cos ^2}{55^0} + {\cos ^2}{65^0} + {\cos ^2}{75^0}\\
 = \left( {{{\sin }^2}{{75}^0} + {{\cos }^2}{{75}^0}} \right) + \left( {{{\sin }^2}{{65}^0} + {{\cos }^2}{{65}^0}} \right)\\
 + \left( {{{\sin }^2}{{55}^0} + {{\cos }^2}{{55}^0}} \right) + \dfrac{1}{2}\\
 = 1 + 1 + 1 + \dfrac{1}{2}\\
 = \dfrac{7}{2}\\
B = {\sin ^2}{10^0} – {\sin ^2}{20^0} + {\sin ^2}{30^0} – {\sin ^2}{40^0}\\
 – si{n^2}{50^0} – {\sin ^2}{70^0} + {\sin ^2}{80^0}\\
 = {\cos ^2}{80^0} + {\sin ^2}{80^0} – \left( {{{\sin }^2}{{20}^0} + {{\sin }^2}{{70}^0}} \right)\\
 – \left( {{{\sin }^2}{{40}^0} + {{\sin }^2}{{50}^0}} \right) + {\sin ^2}{30^0}\\
 = 1 – 1 – 1 + {\left( {\dfrac{1}{2}} \right)^2}\\
 =  – 1 + \dfrac{1}{4}\\
 =  – \dfrac{3}{4}
\end{array}$