Question

【高一数学】化简

1-cos^6（α）-sin^6(α)/1-cos^4(α)-sin^4(α)

Answer 1

(1-cos^6α-sin^6α)/(1-cos^4α-sin^4α)

1=sin^2α+cos^2α
原式=(sin^2α+cos^2α-cos^6α-sin^6α) /(sin^2α+cos^2α-cos^4α-sin^4α)
分母=sin^2α(1-sin^2α)+cos^2α(1-cos^2α)=2sin^2αcos^2α
分子=sin^2α(1-sin^4α)+cos^2α(1-cos^4α)
sin^2α(1-sin^4α)=sin^2α(1-sin^2α)(1+sin^2α)=sin^2αcos^2α(1+sin^2α)
同理cos^2α(1-cos^4α)=sin^2αcos^2α(1+cos^2α)

原式=sin^2αcos^2α(1+sin^2α+1+cos^2α)/2sin^2αcos^2α=3/2

追问

哇靠 神速度啊··

Answer 2

[(sinα)^2+(cosα)^2]^3=(sinα)^6+3(sinα)^4(cosα)^2+3(sinα)^2(cosα)^4+(cosα)^6
所以分子就是:3(sinα)^4(cosα)^2+3(sinα)^2(cosα)^4=3[(sinα)^2(cosα)^2]
分母:2[(sinα)^2(cosα)^2]
所以原式=3/2