solve functional equation: $[f(x)]^2-[f(y)]^2$=$f(x+y)f(x-y)$

i am trying to solve following problems and please guys help me suppose that,there is given following equation $[f(x)]^2-[f(y)]^2$=$f(x+y) \cdot f(x-y)$ there was said that,it requires some knowledge of calculus,first of all i factor this equation as $(f(x)+f(y)) \cdot (f(x)-f(y))$=$f(x+y)\cdot f(x-y)$ so it means that

1.$f(x)+f(y)=f(x+y)$ 2.$f(x)-f(y)=f(x-y)$ so it means that $f(x)=a \cdot x$ right yes?where does it requires calculus?range of x,y are all real numbers

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$f \ast f \neq |f|^2$ – plusepsilon.de Apr 13 '12 at 13:02
You could use "\cdot" instead of "$*$" – Beni Bogosel Apr 13 '12 at 15:25
If two products are equal it doesn't mean the terms are equal. Your conclusions 1) and 2) are not right... – N. S. Apr 19 '12 at 12:14

That is called the Sine Functional Equation; for the beginning you may check this page. It mentions that $f(x)=kx$ satisfies this equation. And of course $\sin^2(x)-\sin^2(y)=\sin(x+y)\cdot \sin(x-y)$