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Giải bài 9 trang 11 sách bài tập toán 11 - Cánh diều

Chứng minh rằng:

⛄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

Chứng minh rằng:

a)    \({\sin ^4}x + {\cos ^4}x = 1 - 2{\sin ^2}x{\cos ^2}x\).

b)    \({\sin ^6}x + {\cos ^6}x = 1 - 3{\sin ^2}x{\cos ^2}x\).

Phương pháp giải - Xem chi tiết

a) 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 \({\sin ^2}x + {\cos ^2}x = 1\). b) Sử dụng hằng đẳng thức \({\left( {A + B} \right)^3} = {A^3} + {B^3} + 3AB\left( {A + B} \right)\) với \(A = {\sin ^2}x\), \(B = {\cos ^2}x\); Sử dụng công thức \({\sin ^2}x + {\cos ^2}x = 1\).

Lời giải chi tiết

a) Ta có: \({\left( {{{\sin }^2}x + {{\cos }^2}x} \right)^2} = {\left( {{{\sin }^2}x} \right)^2} + {\left( {{{\cos }^2}x} \right)^2} + 2{\sin ^2}x{\cos ^2}x\)\( = {\sin ^4}x + {\cos ^4}x + 2{\sin ^2}x{\cos ^2}x\)Do \({\sin ^2}x + {\cos ^2}x = 1\), ta suy ra\({1^2} = {\sin ^4}x + {\cos ^4}x + 2{\sin ^2}x{\cos ^2}x \Rightarrow {\sin ^4}x + {\cos ^4}x = 1 - 2{\sin ^2}x{\cos ^2}x\)Bài toán được chứng minh.b) Ta có: \({\left( {{{\sin }^2}x + {{\cos }^2}x} \right)^3} = {\left( {{{\sin }^2}x} \right)^3} + {\left( {{{\cos }^2}x} \right)^3} + 3{\sin ^2}x{\cos ^2}x\left( {{{\sin }^2}x + {{\cos }^2}x} \right)\)\( = {\sin ^6}x + {\cos ^6}x + 3{\sin ^2}x{\cos ^2}x\left( {{{\sin }^2}x + {{\cos }^2}x} \right)\)Do \({\sin ^2}x + {\cos ^2}x = 1\), ta suy ra\(1 = {\sin ^6}x + {\cos ^6}x + 3{\sin ^2}x{\cos ^2}x \Rightarrow {\sin ^6}x + {\cos ^6}x = 1 - 3{\sin ^2}x{\cos ^2}x\)Bài toán được chứng minh.

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Tham Gia Group Dành Cho Lớp 11 Chia Sẻ, Trao Đổi Tài Liệu Miễn Phí

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