Question

求值：(1-sin^6 x-cos^6 x)/(1-sin^4 x-cos^4 x)

Answer 1

因为sin^4 x+cos^4 x=（sin^2 x+cos^2 x）^2-2sin^2 x cos^2 x=1-2sin^2 x cos^2 x

所以1-sin^4 x-cos^4 x=2sin^2 x cos^2 x

因为

sin^6 x+cos^6 x

=（sin^2 x+cos^2 x）^3-3sin^4 x cos^2 x-3sin^2 x cos^4 x

=1-3sin^2 x cos^2 x（sin^2 x+cos^2 x）

=1-3sin^2 x cos^2 x

所以1-sin^6 x-cos^6 x=3sin^2 x cos^2 x

所以(1-sin^6 x-cos^6 x)/(1-sin^4 x-cos^4 x) =3sin^2 x cos^2 x/（2sin^2 x cos^2 x）=3/2