f(x)=（1+cotx)sin^2x-2sin(x+π/4)sin(x-π/4)

f(x)=(1+cotx)sin²x-2sin(x+π/4)sin(x-π/4)
=sin²x+cotxsin²x+cos2x-cos(π/2)
=sin²x+sinxcosx+cos2x
=1/2*(2sin²x-1+2sinxcosx+1)+cos2x
=1/2*(sin2x-cos2x)+cos2x
=1/2*(sin2x+cos2x)
=√2/2*sin(2x+π/4)
T=2π/2=π
x∈[π/12,π/2]
2x∈[π/6,π]
2x+π/4∈[5π/12,5π/4]
sin(2x+π/4)∈[√2/2,1]
√2/2*sin(2x+π/4)∈[1/2,√2/2]
∴f(x)∈[1/2,√2/2]