f(x)=2sinxcosx+cos2x=sin2x+cos2x=(√2)sin(2x+π/4)
∴f(π/2)=(√2)sin(π+π/4)=1
又∵x∈[-π/2,π/2]
∴x∈[-3π/8,π/8],f(x)单调递增
∴x∈[kπ-3π/8,kπ+π/8],k∈Z,f(x)单调递增