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Giải bài 18 trang 15 sách bài tập toán 11 - Cánh diều

Chọn đẳng thức đúng trong các đẳng thức sau:

💝Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Cánh diều

Toán - Văn - Anh - Lí - Hóa - Sinh
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Đề bài

Chọn đẳng thức đúng trong các đẳng thức sau:

A.   \({\sin ^4}x + {\cos ^4}x = \frac{{3 - \cos 4x}}{4}\)

B.   ♊\({\sin ^4}x + {\cos ^4}x = \frac{{3 + \cos 4x}}{4}\)

C.   🥃\({\sin ^4}x + {\cos ^4}x = \frac{{3 + \cos 4x}}{2}\)

D.   🥀\({\sin ^4}x + {\cos ^4}x = \frac{{3 - \cos 4x}}{2}\)

Phương pháp giải - Xem chi tiết

Sử dụng hằng đẳng thức \({\left( {A + B} \right)^2} = {A^2} + 2AB + {B^2}\) với \(A = {\sin ^2}x\), \(B = {\cos ^2}x\). Sử dụng công thức \(\sin 2x = 2\sin x.\cos x\), \(\cos 2x = 1 - 2{\sin ^2}x\)

Lời giải chi tiết

Ta có:\(\begin{array}{l}{\left( {{{\sin }^2}x + {{\cos }^2}x} \right)^2} = {\left( {{{\sin }^2}x} \right)^2} + {\left( {{{\cos }^2}x} \right)^2} + 2{\sin ^2}x{\cos ^2}x\\ \Rightarrow 1 = {\sin ^4}x + {\cos ^4}x + \frac{1}{2}{\left( {2\sin x.\cos x} \right)^2}\\ \Rightarrow 1 = {\sin ^4}x + {\cos ^4}x + \frac{1}{2}{\sin ^2}2x\\ \Rightarrow {\sin ^4}x + {\cos ^4}x = 1 - \frac{1}{2}{\sin ^2}2x\end{array}\)Mặt khác, ta có \(\cos 4x = 1 - 2{\sin ^2}2x \Rightarrow {\sin ^2}2x = \frac{{1 - \cos 4x}}{2}\)Suy ra \({\sin ^4}x + {\cos ^4}x = 1 - \frac{1}{2}.\frac{{1 - \cos 4x}}{2} = \frac{{3 + \cos 4x}}{4}\)Đáp án đúng là B.

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