Rút gọn biểu thức \(A = {\cos ^2}\left( \alpha \right) + {\cos ^2}\left( {\alpha + \beta } \right) - 2\cos \left( \alpha \right)\cos \left( \beta \right)\cos \left( {\alpha + \beta } \right)\) ta được kết quả
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A.
\({\cos ^2}\left( \alpha \right)\)
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B.
\({\cos ^2}\left( \beta \right)\)
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C.
\({\sin ^2}\left( \alpha \right)\)
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D.
\({\sin ^2}\left( \beta \right)\)
Sử dụng các công thức:
\(\begin{array}{l}\cos \left( \alpha \right) + \cos \left( \beta \right) = 2\cos \left( {\frac{{\alpha + \beta }}{2}} \right)\cos \left( {\frac{{\alpha - \beta }}{2}} \right)\\\sin \left( \alpha \right)\cos \left( \beta \right) = \frac{1}{2}\left[ {\sin \left( {\alpha + \beta } \right) + \sin \left( {\alpha - \beta } \right)} \right]\\\sin \left( {2\alpha } \right) = 2\sin \left( \alpha \right)\cos \left( \beta \right)\\\cos \left( {2\alpha } \right) = 2{\cos ^2}\left( \alpha \right) - 1\end{array}\)
\(\begin{array}{l}A = {\cos ^2}\left( \alpha \right) + {\cos ^2}\left( {\alpha + \beta } \right) - 2\cos \left( \alpha \right)\cos \left( \beta \right)\cos \left( {\alpha + \beta } \right)\\ = {\cos ^2}\left( \alpha \right) + {\cos ^2}\left( {\alpha + \beta } \right) - 2.\frac{1}{2}\left[ {\cos \left( {\alpha + \beta } \right) + \cos \left( {\alpha - \beta } \right)} \right]\cos \left( {\alpha + \beta } \right)\\ = {\cos ^2}\left( \alpha \right) + {\cos ^2}\left( {\alpha + \beta } \right) - {\cos ^2}\left( {\alpha + \beta } \right) - \cos \left( {\alpha - \beta } \right)\cos \left( {\alpha + \beta } \right)\\ = \frac{{1 + \cos \left( {2\alpha } \right)}}{2} - \frac{1}{2}\left[ {\cos \left( {2\alpha } \right) + \cos \left( {2\beta } \right)} \right]\\ = \frac{1}{2} - \frac{{\cos \left( {2\beta } \right)}}{2} = {\sin ^2}\left( \beta \right)\end{array}\)
Đáp án : D
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