1.若sinα+cosα=1/3,则cos4α的值等于____
解析：∵sinα+cosα=1/3
∴sinα=1/3-cosα==> (sinα)^2=1/9+(cosα)^2-2/3cosα
代入(sinα)^2+(cosα)^2=1
得9(cosα)^2-3(cosα)-4=0==>cosα=(1-√17)/6或cosα=(1+√17)/6
∴cosα=(1-√17)/6,sinα=(1+√17)/6或cosα=(1+√17)/6,sinα=(1-√17)/6
∴cos4α=2[(cosα-sinα)(cosα+sinα)]^2-1
=2*(√17/3*1/3)^2-1=-47/81
2.已知函数f（x）=2cosxcos（π/6-x）-根号3*sin^2 x+sinxcosx,求f（x）的最小正周期
解析：∵函数f(x)=2cosxcos(π/6-x)-√3(sinx)^2+sinxcosx
=√3(sinx)^2+sinxcosx-√3(sinx)^2+sinxcosx
=√3cos2x+sin2x=2sin(2x+π/3)
∴T=2π/2=π
f（x）的最小正周期为π