若sinα+ cosα=m,求sin2α-cos4α的值
问题描述：
若sinα+ cosα=m,求sin2α-cos4α的值
答
2m^4-3m^2
答
sina+cosa=m
平方得1+2sinacosa=m^2
则sin2a=2sinacosa=m^2-1
cos4a=1-2sin2a^2=1-2(m^2-1)^2
所以结果为
m^2-1-[1-2(m^2-1)^2]
=m^2-2+2(m^2-1)^2
=2m^4-3m^2
答
sinα+cosα=m
(sinα+cosα)^2=m^2
(sinα)^2+(cosα)^2+2sinαcosα=m^2
1+sin2α=m^2
sin2α=m^2-1
sin2α-cos4α
=sin2α-(1-2(sin2α)^2)
=2(sin2α)^2+sin2α-1
=2(m^2-1)^2+(m^2-1)-1
=2m^4-4m^2+2+m^2-1-1
=2m^4-3m^2