函数f(x)=sin²(x+π/4)-sin²(x-π/4)是奇函数还是偶函数,还有周期是多少,请把化简的过程写写

函数f(x)=sin²(x+π/4)-sin²(x-π/4)是奇函数还是偶函数,还有周期是多少,请把化简的过程写写
函数f(x)=sin²(x+π/4)-sin²(x-π/4)是奇函数还是偶函数,还有周期是多少,请把化简的过程写写

函数f(x)=sin²(x+π/4)-sin²(x-π/4)是奇函数还是偶函数,还有周期是多少,请把化简的过程写写
f(x)=sin²(x+π/4)-sin²(x-π/4) (先降次)
=1/2[1-cos(2x+π/2)]-1/2[1-cos(2x-π/2)]
=-1/2[cos(2x+π/2)-cos(2x-π/2)]
=-1/2[-sin(2x)-sin(2x)]
=sin(2x)
所以是奇函数,周期π