cos（X+π/4）+cos（X-π/4）=

【解1】用【和差化积】公式cosα+cosβ=2cos[(α+β)/2]cos[(α-β)/2]当α=x+π/4，β=x+π/4时，α+β=2x，α-β=π/2∴ cos(x+π/4)+cos(x-π/4)=2cosxcos(π/4)=(√2)cosx【解2】用公式cos(α+β)=cosαcosβ-sinαsinβ原式=[cosxcos(π/4)-sinxsin(π/4)]+[cosxcos(π/4)+sinxsin(π/4)]=2cosxcos(π/4)=(√2)cosx【解3】用公式 cosα=sin(π/2-α)和cosx=cos(-α)原式=sin(π/4-x)+cos(π/4-x)=(√2)[sin(π/4-x)cos(π/4)+cos(π/4-x)sin(π/4)]=(√2)sin[(π/4-x)+π/4]=(√2)sin(π/2-x)=√2cosx

cos(x+π/4)=-sin(x-π/4)是对的。解：因为cos(x+π/4)=cosxcosπ/4-sinxsinπ/4=√2/2*cosx-√2/2*sinx=-(√2/2*sinx-√2/2*cosx)，而sin(x-π/4)=sinxcosπ/4-cosxsinπ/4=√2/2*sinx-√2/2*cosx所以可知cos(x+π/4)=-sin(x-π/4)。扩展资料：1、三角函数二角和差公式（1）sin(A+B)=sinAcosB+cosAsinB（2）sin(A-B)=sinAcosB-cosAsinB（3）cos(A+B)=cosAcosB-sinAsinB（4）cos(A-B)=cosAcosB+sinAsinB2、三角函数积化和差公式（1）cosAcosB=1/2*(cos(A+B)+cos(A-B))（2）sinAsinB=1/2*(cos(A-B)-cos(A+B))（3）cosAsinB=1/2*(sin(A+B)-sin(A-B))（4）sinAcosB=1/2*(sin(A+B)+sin(A-B))3、特殊角的三角函数值sinπ/6=1/2、cosπ/6=√3/2、tanπ/6=√3/3、cotπ/6=√3sinπ/4=√2/2、cosπ/4=√2/2、tanπ/4=1、cotπ/4=1、sinπ/3=√3/2、cosπ/3=1/2、tanπ/3=√3、cotπ/3=√3/3sinπ/2=1、cosπ/2=0、tanπ/2不存在、cotπ/2=0

cos(x+π/4)+cos(x-π/4)=cosx*cos(π/4)-sinx*sin(π/4)+cosx*cos(π/4)+sinx*sin(π/4)=2cosx*cos(π/4)=2cosx*√2/2=√2cosx