题目

用cosα表示sin四次方α-sin²α+cos²α
求证2(1-sinα)(1+cosα)=(1-sinα+cosα)²
求证sin²α+sin²β-sin²αsin²β+cos²αcos²β=1

提问时间：2021-01-30

答案

sin^4α-sin^2α+cos^2α
=sin^2α(sin^2α-1)+cos^2α
=-sin^2αcos^2α+cos^2α
=cos^2α(1-sin^2α)
=cos^4α
2(1-sinα)(1+cosα)
=2(1+cosα-sinα-sinαcosα)
=2+2cosα-2sinα-2sinαcosα
=(sinα-cosα)^2-2(sinα-cosα)+1
=(sinα-cosα-1)^2
=(1-sinα+cosα)^2
sin^2α+sin^2β-sin^αsin^2β+cos^2αcos^2β
=sin^2α(1-sin^2β)+sin^2β+cos^2αcos^2β
=sin^2αcos^2β+sin^2β+cos^2αcos^2β
=cos^2β(sin^2α+cos^2α)+sin^2β
=cos^2β+sin^2β
=1