作业帮cos^6(π/8)-sin^6(π/8)=求值,
来源：学生作业帮 编辑：拍题作业网作业帮 分类：数学作业 时间：2024/05/22 05:27:13
cos^6(π/8)-sin^6(π/8)=求值,
[cos^2(π/8)-sin^2(π/8)][cos^4(π/8)+sin^4(π/8)+cos^2(π/8)sin^2(π/8)]
=cos(π/4)[1-cos^2(π/8)sin^2(π/8)]
=cos(π/4)[1-[sin^2(π/4)]/4]
==7√2/16
[cos^2(π/8)-sin^2(π/8)][cos^4(π/8)+sin^4(π/8)+cos^2(π/8)sin^2(π/8)]
=cos(π/4)[1-cos^2(π/8)sin^2(π/8)]
=cos(π/4)[1-[sin^2(π/4)]/4]
==7√2/16
知识储备：a³ +b³ = (a+b)(a²-ab +b²)
a³ -b³= (a-b)(a² +ab +b²)
cos^6(π/8)-sin^6(π/8)=[cos³(π/8)-sin³(π/8)]·[cos³(π/8)+sin³(π/8)]
={[cos(π/8)-sin(π/8)][cos²(π/8) +cos(π/8)·sin(π/8)+sin²(π/8)]}· {[cos(π/8)+sin(π/8)][cos²(π/8) -cos(π/8)·sin(π/8)+sin²(π/8)]}
=[cos²(π/8)-sin²(π/8)]·[1+cos(π/8)·sin(π/8)]·[1-cos(π/8)·sin(π/8)]
=cos(π/4)·[1+(1/2)sin(π/4)]·[1-(1/2)sin(π/4)
=(√2/2)·[1+(√2/4)]·[1-(√2/4)]
=7√2/16
a³ -b³= (a-b)(a² +ab +b²)
cos^6(π/8)-sin^6(π/8)=[cos³(π/8)-sin³(π/8)]·[cos³(π/8)+sin³(π/8)]
={[cos(π/8)-sin(π/8)][cos²(π/8) +cos(π/8)·sin(π/8)+sin²(π/8)]}· {[cos(π/8)+sin(π/8)][cos²(π/8) -cos(π/8)·sin(π/8)+sin²(π/8)]}
=[cos²(π/8)-sin²(π/8)]·[1+cos(π/8)·sin(π/8)]·[1-cos(π/8)·sin(π/8)]
=cos(π/4)·[1+(1/2)sin(π/4)]·[1-(1/2)sin(π/4)
=(√2/2)·[1+(√2/4)]·[1-(√2/4)]
=7√2/16
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