函数 (4 20:28:50)cos^4 (π /8)+cos^4(3π/8)+cos^4(5π/8)+cos^4(7π/8) =

￥表示派 cos^4(￥7/8)=cos^4(￥-￥/8)=cos^4(￥/8) 同理可得cos^4(3￥/8)=cos^4(5￥/8) 原式等于2cos^4(￥/8)+2cos^4(3￥/8) cos^4(3￥/8)=cos^4(￥/2-￥/8)=sin^4(￥/8) 原式等于2cos^4(￥/8)+2sin^4(￥/8) 又通过sin^2+cos^2=1配方得到2sin^4(x)+2cos^4(x)=2-4cos^2(x)sin^2(x) 原式等于2-4cos^2(￥/8)sin^2(￥/8)=2-sin^2(￥/4)=2-1/2=3/2

2cos^2 (x)-1=cos 2x 4cos^4 (x)=(cos 2x+1)² 所以
cos^4 (π /8)+cos^4(3π/8)+cos^4(5π/8)+cos^4(7π/8)
=0.25[4cos^4 (π /8)+4cos^4(3π/8)+4cos^4(5π/8)+4cos^4(7π/8)]
=0.25[(cosπ/4+1)² +(cos3π/4+1)²+(cos5π/4+1)²+(cos7π/4+1)² ]
=0.25[2(根号2/2+1)²+2(-根号2/2+1)²]
=0.25×6
=1.5

cos^4 (π /8)+cos^4(3π/8)+cos^4(5π/8)+cos^4(7π/8) =
先用诱导公式进行转换
=cos^4 (π /8)+cos^4(3π/8)+cos^4(π-3π/8)+cos^4(π-π/8)
=cos^4 (π /8)+cos^4(3π/8)+cos^4(3π/8)+cos^4(π/8)
=2cos^4(π /8)+2cos^4(3π/8)
=2cos^4(π/8)+2cos^4(π/2-π/8)
=2cos^4(π/8)+2sin^4(π/8)
=2[sin²(π/8)+cos²(π/8)]²-4sin²(π/8)cos²(π/8)
=2-sin²π/4=3/2
其实有些步骤可以跳过不些~不过为了清楚些,我还是都写上了