求证3+cos4α-4cos2α=(8sinα)^4

来源：学生作业帮编辑：百度作业网作业帮分类：数学作业时间：2024/07/16 11:15:06

求证3+cos4α-4cos2α=(8sinα)^4
证（1-cos2α)/(1+cos2α)=tan²α
(1+sin2α)/(cosα+sinα)=cosα+sinα

证明：1）3+cos4α-4cos2α=2+(1+cos4α)-4cos2α
=2+2(cos2α)^2-4cos2α
=2[1+(cos2α)^2-2cos2α]
=2(1-cos2α)^2
=2*{1-[(cosα)^2-(sinα)^2}^2
=2*{1-(cosα)^2+(sinα)^2}^2
=2*{2(sinα)^2}^2
=8*(sinα)^4
2）（1-cos2α)/(1+cos2α)={1-[(cosα)^2-(sinα)^2]}/{1+[(cosα)^2-(sinα)^2]}
={1-(cosα)^2+(sinα)^2}/{1-(sinα)^2+(cosα)^2}
=2(sinα)^2/{2(cosα)^2}
=tan²α
3）(1+sin2α)/(cosα+sinα)=(sin²α+cos²α+2sinαcosα)/(cosα+sinα)
=(cosα+sinα)^2/(cosα+sinα)
=cosα+sinα

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