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Câu hỏi: Trong các hệ thức sau, hệ thức nào không đúng ?
(A) \({(\sin \alpha + \cos \alpha)^2} = 1 + 2\sin \alpha \cos \alpha \);
(B) \({(\sin \alpha - \cos \alpha)^2} = 1 - 2\sin \alpha \cos \alpha \);
(C) \({\cos ^4}\alpha - {\sin ^4}\alpha = {\cos ^2}\alpha - {\sin ^2}\alpha \);
(D) \({\cos ^4}\alpha + {\sin ^4}\alpha = 1\).
(A) \({(\sin \alpha + \cos \alpha)^2} = 1 + 2\sin \alpha \cos \alpha \);
(B) \({(\sin \alpha - \cos \alpha)^2} = 1 - 2\sin \alpha \cos \alpha \);
(C) \({\cos ^4}\alpha - {\sin ^4}\alpha = {\cos ^2}\alpha - {\sin ^2}\alpha \);
(D) \({\cos ^4}\alpha + {\sin ^4}\alpha = 1\).
Lời giải chi tiết
Ta có:
\(\begin{array}{l}
{\left({\sin \alpha + \cos \alpha } \right)^2}\\
= {\sin ^2}\alpha + 2\sin \alpha \cos \alpha + {\cos ^2}\alpha \\
= \left({{{\sin }^2}\alpha + {{\cos }^2}\alpha } \right) + 2\sin \alpha \cos \alpha \\
= 1 + 2\sin \alpha \cos \alpha
\end{array}\)
nên A đúng.
\(\begin{array}{l}
{\left({\sin \alpha - \cos \alpha } \right)^2}\\
= {\sin ^2}\alpha - 2\sin \alpha \cos \alpha + {\cos ^2}\alpha \\
= \left({{{\sin }^2}\alpha + {{\cos }^2}\alpha } \right) - 2\sin \alpha \cos \alpha \\
= 1 - 2\sin \alpha \cos \alpha
\end{array}\)
nên B đúng.
\(\begin{array}{l}
{\cos ^4}\alpha - {\sin ^4}\alpha \\
= \left({{{\cos }^2}\alpha - {{\sin }^2}\alpha } \right)\left({{{\cos }^2}\alpha + {{\sin }^2}\alpha } \right)\\
= \left({{{\cos }^2}\alpha - {{\sin }^2}\alpha } \right). 1\\
= {\cos ^2}\alpha - {\sin ^2}\alpha
\end{array}\)
nên C đúng.
\({\cos ^4}\alpha + {\sin ^4}\alpha \)
\(= {({\sin ^2}\alpha + {\cos ^2}\alpha)^2} - 2{\sin ^2}\alpha {\cos ^2}\alpha \)
\(= 1 - 2{\sin ^2}\alpha {\cos ^2}\alpha. \)
nên D sai.
Chọn (D).
Ta có:
\(\begin{array}{l}
{\left({\sin \alpha + \cos \alpha } \right)^2}\\
= {\sin ^2}\alpha + 2\sin \alpha \cos \alpha + {\cos ^2}\alpha \\
= \left({{{\sin }^2}\alpha + {{\cos }^2}\alpha } \right) + 2\sin \alpha \cos \alpha \\
= 1 + 2\sin \alpha \cos \alpha
\end{array}\)
nên A đúng.
\(\begin{array}{l}
{\left({\sin \alpha - \cos \alpha } \right)^2}\\
= {\sin ^2}\alpha - 2\sin \alpha \cos \alpha + {\cos ^2}\alpha \\
= \left({{{\sin }^2}\alpha + {{\cos }^2}\alpha } \right) - 2\sin \alpha \cos \alpha \\
= 1 - 2\sin \alpha \cos \alpha
\end{array}\)
nên B đúng.
\(\begin{array}{l}
{\cos ^4}\alpha - {\sin ^4}\alpha \\
= \left({{{\cos }^2}\alpha - {{\sin }^2}\alpha } \right)\left({{{\cos }^2}\alpha + {{\sin }^2}\alpha } \right)\\
= \left({{{\cos }^2}\alpha - {{\sin }^2}\alpha } \right). 1\\
= {\cos ^2}\alpha - {\sin ^2}\alpha
\end{array}\)
nên C đúng.
\({\cos ^4}\alpha + {\sin ^4}\alpha \)
\(= {({\sin ^2}\alpha + {\cos ^2}\alpha)^2} - 2{\sin ^2}\alpha {\cos ^2}\alpha \)
\(= 1 - 2{\sin ^2}\alpha {\cos ^2}\alpha. \)
nên D sai.
Chọn (D).