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
That is called the sine functional equation; for a start you may check this. It mentions that $f(x)=kx$ satisfies the equation. Of course, we also have $$ \sin^2(x)-\sin^2(y)=\sin(x+y)\cdot \sin(x-y). $$