(1)
f（x）=2cos（x+π/4）cos（x-π/4）=2sin(x-π/4)*cos(x-π/4)=sin(2x-π/2)=cos(2x)
x∈（2kπ,2kπ+π/2）,k∈Z时,f(x)单调递增
(2)f（α）=sin(2α-π/2)=12/13,α∈（-π/2,0）,
2α-π/2∈(-3π/2,-π),
cos(2α-π/2)=-5/13
α-π/4∈(-3π/4,-π/2),
t=sin（α-π/4）<0
1-2t^2=-5/13
t=sin（α-π/4）=-3*(√13)/13
(3)
g（x）=f（x）+√3sin2x=2[cos(2x)/2+√3sin(2x)/2]=sin(2x+π/6),x∈[-π/6,π/4],
g(π/4)=(√3)/2
g(-π/6)=-1/2
g(x)min=-1/2
g(π/6)=g(x)max=1