已知函数f(x)=[2cos^4x-2cos^x+1/2]/[tan(π/4-x)sin^2(π/4+x)],求f(x)的值域

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已知函数f(x)=[2cos^4x-2cos^x+1/2]/[tan(π/4-x)sin^2(π/4+x)],求f(x)的值域

已知函数f(x)=[2cos^4x-2cos^x+1/2]/[tan(π/4-x)sin^2(π/4+x)],求f(x)的值域
已知函数f(x)=[2cos^4x-2cos^x+1/2]/[tan(π/4-x)sin^2(π/4+x)],求f(x)的值域

已知函数f(x)=[2cos^4x-2cos^x+1/2]/[tan(π/4-x)sin^2(π/4+x)],求f(x)的值域
tan(π/4-x)= (tanπ/4-tanx)/(1+tanπ/4tanx) = (1-tanx)(1+tanx) = (cosx-sinx)/(cosx+sinx)
sin^2(π/4+x) = (√2/2sinx+√2/2cosx)^2= 1/2(sinx+cosx)^2
f(x) = [2cos^4x-2cos^2x+1/2]/[tan(π/4-x)sin^2(π/4+x)]
=[2cos^4x-2cos^2x+1/2]/[ (cosx-sinx)/(cosx+sinx)*1/2(sin+cosx)^2]
=[4cos^4x-4cos^2x+1]/[ (cosx-sinx)(cosx+sinx)]
= (2cos^2x-1)^2 / (cos^2x-sin^2x)
=[cos(2x)]^2 / cos(2x)
= cos(2x) ∈【-1,1】
值域:【-1,1】