Question

已知cos2X=根号2/3，则sin^4X+cos^4的值为？求详细过程！

Answer 1

y=sin^4x+cos^4x
=sin^4x+2sin^2xcos^2x+cos^4x-2sin^2xcos^2x
=(sin^2 2x+cos^2 2x)^2-(1/2)(2sinxcosx)^2
=1-(1/2)(sin2x)^2
=1-(1/2)(1-cos^2 2x)
=1-1/2*(1-2/9)
=1-7/18
=11/18

追问

sin^4x+2sin^2xcos^2x+cos^4x-2sin^2xcos^2x
怎么得到=(sin^2 2x+cos^2 2x)^2-(1/2)(2sinxcosx)^2
怎么X变成2X??

追答

sin^4x+2sin^2xcos^2x+cos^4x=[(sin^2（2x)+cos^2( 2x)]^2

(1/2)(2sinxcosx)^2=1/2*4*sin^2xcos^2x=2sin^2xcos^2x

明白了吗

Answer 2

sin^4X+cos^4x
=(cos^2x+sin^2x)^2-1/2(2sinxcosx)^2
=1-1/2sin^2(2x)
=1-1/2[1-cos^2(2x)]
=1-1/2*[1-(√ 2/3)^2]
=1-1/2(1-2/9)
=1-1/2*7/9
=1-7/18
=11/18