已知函数f(x)=cos(2x-π/3)+2sin(x-π/4)sin(x+π/2) 化简为sin的形式

题目

已知函数f(x)=cos(2x-π/3)+2sin(x-π/4)sin(x+π/2) 化简为sin的形式

答案

利用三角函数积化和差及和差化积公式,
f(x)=cos(2x-π/3）-[cos(x-π/4+x+π/2)-cos(x-π/4-x-π/2)]
=cos(2x-π/3）-[cos(2x+π/4)-cos(-3π/4)]
=cos(2x-π/3）-cos(2x+π/4)+cosπ/4
=-2sin[(2x-π/3+2x+π/4)/2]sin[(2x-π/3-2x-π/4)/2]+√2/2
=-2sin(2x-π/24)sin(-7π/24)+√2/2
=2sin(2x-π/24）sin(7π/24)+√2/2.