Characterization of sine and cosine functions. Uniqueness of $\pi$.

I would like to prove the following theorem:

There only exists two functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ which satisfies the following properties:

1. For all $x\in\mathbb{R}$, $f^2(x)+g^2(x)=1$.
2. For all $x,y\in\mathbb{R}$, $f(x+y)=f(x)\cdot g(y)+g(x)\cdot f(y)$.
3. For all $x,y\in\mathbb{R}$, $g(x+y)=g(x)\cdot g(y)-f(x)\cdot f(y)$.
4. There exists $\gamma>0$ such that $0<f(x)<x<\dfrac{f(x)}{g(x)}$, for all $x\in(0,\gamma)$.

The proof goes as follows:

Suppose that f and g exist and verify the properties the theorem mentions. If we got explicit formulas for those functions which verify the properties of the theorem, we proved the existence and uniqueness at the same time.

We are going to prove a sequence of properties that those function would verify:

1. For all $x\in\mathbb{R}$, $-1\leq f(x),g(x)\leq 1$.
2. $f(0)=0$ and $g(0)=1$.
3. $f$ is odd and $g$ is even.
4. For all $x,y\in\mathbb{R}$, $f(x)\pm f(y)=2\cdot f(\dfrac{x\pm y}{2})\cdot g(\dfrac{x\mp y}{2})$.
5. $f$ is continuous.
6. For all $x,y\in\mathbb{R}$, $g(x)+g(y)=2\cdot g(\dfrac{x+y}{2})\cdot g(\dfrac{x-y}{2})$.
7. For all $x,y\in\mathbb{R}$, $g(x)-g(y)=-2\cdot f(\dfrac{x+y}{2})\cdot f(\dfrac{x-y}{2})$.
8. $g$ is continuous.

This is the property where I got stuck.

9. There exists $x\in\mathbb{R}$ such that $g(x)=0$. Furthermore, $x$ does not depend on $(f,g)$, i.e., if $f_1,f_2,g_1,g_2:\mathbb{R}\rightarrow\mathbb{R}$ are functions verifying the conditions of the theorem and $x\in\mathbb{R}$ is such that $g_1(x)=0$, then $g_2(x)=0$. We will define $\pi:=2\cdot\min\{x>0:g(x)=0\}$.

You define $\beta:=\inf\{g(x):x>0\}$. If $\beta>0$ you get a contradiction with $f$ not being upper bounded. Using continuity, you prove the existence of $x=x(g)$. How can I prove the independe of $x$ on $g$?

Proved what's above, it is very easy to see that $f$ and $g$ are uniquely defined for $x=\dfrac{m\cdot\pi}{2^n}$ with $m\in\mathbb{Z}$ and $n\in\mathbb{N}\cup\{0\}$. The set $\{\dfrac{m\cdot\pi}{2^n}:m,n\in\mathbb{N}\}$ is dense on $\mathbb{R}$ and, therefore, the extension to $\mathbb{R}$ is unique.

Then we should verify that the properties of the theorem are satisfied with these definitions of $f$ and $g$ to prove existence.

Is there anyone who can help?

Thanks in advance.