Question

Rút gọn biểu thức:

sin⁴\(\alpha\left(1+2\cos^2\alpha\right)+\cos^4\alpha\left(1+sin^2\alpha\right)\)

Answer 1

sữa đề nha : \(sin^4\alpha\left(1+2cos^2\alpha\right)+cos^4\alpha\left(1+2sin^2\alpha\right)\)

ta có : \(sin^4\alpha\left(1+2cos^2\alpha\right)+cos^4\alpha\left(1+2sin^2\alpha\right)\)

\(sin^4\alpha+2cos^2\alpha.sin^4\alpha+cos^4\alpha+2sin^2\alpha.cos^4\alpha\)

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

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

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