答：
f(x)=sin(x+b)+cos(x+b)是奇函数,则f(-x)=-f(x)
f(-x)=sin(-x+b)+cos(-x+b)=-f(x)=-sin(x+b)-cos(x+b)
-sin(x-b)+cos(x-b)=-sin(x+b)-cos(x+b)
-sinxcosb+cosxsinb+cosxcosb+sinxsinb=-sinxcosb-cosxsinb-cosxcosb+sinxsinb
cosxsinb+cosxcosb=0
(sinb+cosb)cosx=0
上式对任意x都成立,则：sinx+cosb=0
所以：b+π/4=2kπ,b=2kπ-π/4
所以：选择B