√2sin(x/2+π/4)cos(x/2)=√2/2sin(x+π/4)+ 1/2 怎么得到的

题目

√2sin(x/2+π/4)cos(x/2)=√2/2sin(x+π/4)+ 1/2 怎么得到的

答案

√2sin(x/2+π/4)cos(x/2)
=√2（sin(x/2)cosπ/4+sinπ/4cos(x/2)）cos(x/2)
=√2(√2/2sin(x/2)+√2/2cos(x/2))cos(x/2)
=sin(x/2)cos(x/2)+cos²(x/2)
=sin2x/2+(1+cos2x)/2
=sin2x/2+cos2x/2+1/2
=(√2/2)[sin2x(√2/2)+cos2x(√2/2)]+1/2
=(√2/2)[sin2xcosπ/4+cos2xsinπ/4]+1/2
=√2/2sin(x+π/4)+ 1/2
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