已知函数f（x）等于sinx加cosx

1、
f(x)=sinx+cosx
f(x)=2f(-x)
∴sinx+cosx=2[sin(-x)+cos(-x)]
sinx+cosx=﹣2sinx+2cosx
3sinx=cosx
tanx=sinx / cosx=1/3
(cos²x-sinxcosx)/(1+sin²x)
=(cos²x-sinxcosx)/(sin²+cos²+sin²x)
=(cos²x-sinxcosx)/(2sin²+cos²) (分式上下同时除以cos²x,得)
=(1 - tanx)/(2tan²x+1)
=6/11
2、
F(x)=f(x)f(-x) + f²(x)
=(sinx+cosx)(-sinx+cosx) + (sinx+cosx)²
=cos²x-sin²x + sin²x+cos²x+2sinxcosx
=2cos²x+2sinxcosx
=cos2x+sin2x+1
=√2[(√2/2)cos2x+(√2/2)sin2x] + 1
=√2sin(2x + π/4) + 1
最大值为√2 + 1
-π/2 + 2kπ≤2x + π/4≤π/2 + 2kπ,k∈Z
-3π/4 + 2kπ≤2x≤π/4 + 2kπ,k∈Z
-3π/8 + kπ≤x≤π/8 + kπ,k∈Z
∴单调递增区间为[-3π/8 + kπ,π/8 + kπ],k∈Z