Verify sin(x+y)cos(x-y)=sin(x)cos(x)+sin(y)cos(y)

Question

Use Identities or formulas to change one side to equal the other.

Answer 1

sin(x+y)=sin(x)cos(y)+cos(x)sin(y); cos(x-y)=cos(x)cos(y)+sin(x)sin(y).

sin(x+y)cos(x-y)=

[sin(x)cos(y)+cos(x)sin(y)][cos(x)cos(y)+sin(x)sin(y)]=

sin(x)cos(x)cos²(y)+sin²(x)cos(y)sin(y)+cos²(x)sin(y)cos(y)+cos(x)sin²(y)sin(x)=

[sin(x)cos(x)cos²(y)+cos(x)sin²(y)sin(x)]+[sin²(x)cos(y)sin(y)+cos²(x)sin(y)cos(y)]=

sin(x)cos(x)(sin²(y)+cos²(y))+sin(y)cos(y)(sin²(x)+cos²(x))=

sin(x)cos(x)+sin(y)cos(y) QED.