Question

[Nguyễn Ánh Tuyền]

Cho \(\alpha\) là góc nhọn. Rút gọn biểu thức:

\(A=sin^6\alpha+cos^6\alpha+3sin^2\alpha-cos^2\alpha\)

cảm ơn các bạn trước nhé

Answer 1

[Hoàng Lê Bảo Ngọc]

\(A=sin^6\alpha+cos^6\alpha+3sin^2\alpha-cos^2\alpha\)

\(=\left(sin^2\alpha\right)^3+\left(cos^2\alpha\right)^3+3sin^2\alpha-cos^2\alpha\)

\(=\left(sin^2\alpha+cos^2\alpha\right)\left(sin^4\alpha+cos^4\alpha-sin^2\alpha.cos^2\alpha\right)+3sin^2\alpha-cos^2\alpha\)

\(=sin^4\alpha+cos^4\alpha-sin^2\alpha.cos^2\alpha+3sin^2\alpha-cos^2\alpha\)

\(=\left(sin^2\alpha+cos^2\alpha\right)^2-2sin^2\alpha.cos^2\alpha-sin^2\alpha.cos^2\alpha+3sin^2\alpha-cos^2\alpha\)

\(1-3sin^2\alpha.cos^2\alpha+3sin^2\alpha-cos^2\alpha=3sin^2\alpha\left(1-cos^2\alpha\right)+\left(1-cos^2\alpha\right)\)

\(=\left(3sin^2\alpha+1\right).sin^2\alpha=0\)

Answer 2

[Lê Quỳnh Trang]