(1)
y=(sinx+a)(cosx+a) = sinxcosx+a(sinx+cosx)+a^2
令t = sinx+cosx
则 t = √2sin(x+π/4)
|t|≤√2
由 (sinx+cosx)^2=(sinx)^2+2sinxcosx+(cosx)^2
所以 t^2 = 1+2sinxcosx
所以 sinxcosx = (t^2-1)/2
则
y = (t^2-1)/2 + at +a^2 = 1/2(t+a)^2 +1/2(a^2 - 1)
因|t|≤√2,常数a>3/2
所以 t = -√2 时,函数y取最小值,即最小值为:
ymin = 1/2(-√2+a)^2 +1/2(a^2-1) = a^2 -√2a +1/2
(2)
设P = cosxsiny
因sin(x+y)=snxcosy+cosxsiny=1/2+P ∈[-1,1]
解得 -3/2≤P≤1/2 (1)式
又 sin(x-y)=sinxcosy-coxsiny=1/2-P∈[-1,1]
解得 -1/2≤P≤3/2 (2)式
取(1)(2)式的交集,得
-1/2≤P≤1/2
所以cosxsiny的最小值是 -1/2