This was part of last week's homework assignment in my Clac 1 class:
Let $f:[0,\frac{\pi}{2}]\rightarrow \mathbb{R}^+\cup\{0\}$ be continuous, $$f(\frac{\pi}{2})=0 \hspace{2mm} \textrm{and} \hspace{2mm} f(x+y)= f(x)f(y)-\sqrt{1-f(x)^2}\sqrt{1-f(y)^2}$$
Show that $f$ is uniquely determined and well-defined.
I tried proving this by first showing that $f(\frac{x}{2})$ , $f(\frac{x}{2^k})$ and in turn $f(\frac{\pi}{2^k})$ and $f(\frac{j\hspace{1mm}\pi}{2^k})$ are uniquely determined. From that I can also see that $\lim_{x\to 0}f(x)=1$ and since $f$ is continuous $f(x)=1$. After this I appear to be stuck tho.
Any hints would be greatly appreciated, keep in mind that I am not allowed to use any advanced (anything beyond 1st semester) theorems.