既知の関数f(x)=[2cos^4x-2cos^x+1/2]/[tan(π/4-x)sin^2(π/4+x)],f(x)の値域を求める

既知の関数f(x)=[2cos^4x-2cos^x+1/2]/[tan(π/4-x)sin^2(π/4+x)],f(x)の値域を求める

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)/(cos^2x-sin^2x)
=[cos(2x)]^2/cos(2x)
=cos(2x)∈[-1,1]
値ドメイン:[-1,1]