Development of circular function through differential equation Suppose there exists two functions $f(x), g(x)$ which satisfy $$f'(x) = g(x), f(0) = 0, g'(x) = -f(x), g(0) = 1$$ for all $x \in \mathbb{R}$. Is it possible to develop the full theory of circular functions $f(x) = \sin x, g(x) = \cos x$ starting with the given differential equations and initial values? In particular I would like to know if it is possible to establish the existence/uniqueness of such functions $f(x), g(x)$ and the fundamental addition theorems $$f(x + y) = f(x)g(y) + g(x)f(y), g(x + y) = g(x)g(y) - f(x)f(y)$$ An answer based on the basic theorems of elementary calculus would be preferred and in particular I would like to avoid any theorems concerning the solution of a first / second order differential equations.
 A: With just differential calculus, I doubt that you'll have much success in proving the existence of solutions of your system of differential equations.  On the other hand, once you do have 
that existence, the uniqueness is easy.
Note that for any solution $(f,g)$ of the system, $f^2 + g^2$ is constant
since 
 $$ \dfrac{d}{dx} \left(f(x)^2 + g(x)^2\right) = 2 f(x) g(x) - 2 g(x) f(x) = 0 $$
The uniqueness follows from this: if $(f_1, g_1)$ and $(f_2, g_2)$ are solutions with the same initial values $f_1(0) = f_2(0)$, $g_1(0) = g_2(0)$, then $f_3 = f_1 - f_2, g_3 = g_1 - g_2$ is also a solution with initial values
$f_3(0) = g_3(0) = 0$, and then $f_3(x)^2 + g_3(x)^2 = 0$ implying $f_3(x) = g_3(x) = 0$.
Then, for example: if $A(y) = f(x+y) - f(x) g(y) - g(x) f(y)$ and $B(y) = g(x+y) -g(x) g(y) + f(x) f(y)$, then $A(0) = B(0) = 0$ and 
$ A'(y) = B(y)$, $B'(y) = -A(y)$.  So again you get $A(y) = 0$, $B(y) = 0$, and these are your addition theorems.
A: I'd suggest to look at Fourier transform for that, or Fourier series if you'd like to stay in the real domain. Specifically, Fourier transform replaces differentiation with multiplication, and that allows you to talk about eigenvalues and eigenvectors of differential operators. After Fourier transform it's an easy exercise to show that eigenspace for your differential equation for a given eigenvalue has dimension 2, and that would prove that $\sin(x),\cos(x)$ span it.
