已知函数f(x)=sin(x+7派/4)+cos(x-3派/4),x

Question

已知函数f(x)=sin(x+7派/4)+cos(x-3派/4),x

Answer 1

题是：已知函数f(x)=sin(x+7π/4)+cos(x-3π/4),x∈R
(1)求f(x)的最小正周期和最小值
(2)已知cos(β-α)=4/5 cos(β+α)=-4/5, 0＜α＜β≤π/2,求证[f（β）]²-2=0

(1)
f(x)=sin(x+7π/4)+cos(x-3π/4)
=sin(x+7π/4)+sin(5π/4-x)
=2sin(3π/2)cos(x+π/4)
=-cos(x+π/4)
最小正周期T=2π/1=2π

(2)
已知cos(β-α)=4/5 cos(β+α)=-4/5,
两式相加 cos(β-α)+cos(β+α)=0
2cosβcosα=0
0＜α＜β≤π/2
cosα≠0
所以cosβ=0
β=π/2
f(β)=f(π/2)=-2cos(π/2+π/4)=2sin(π/4)=√2
所以[f（β）]²-2=(√2)²-2=0