已知函数f(x)=cos^4x-2sinxcosx-sin^4x,求f(x)的值域

题目

已知函数f(x)=cos^4x-2sinxcosx-sin^4x,求f(x)的值域

答案

f(x)=cos^4x-2sinxcosx-sin^4x
=(cos^2x+sin^2x)(cos^2x-sin^2x)-2sinxcosx
=cos2x-sin2x
=√2cos(2x+π/4)
所以值域为[-√2,√2]