Concluding that sine and cosine are $2\pi$ periodic from definitions

Here is a tedious argument that is less slick than Rudin's but perhaps has more geometric content:

The essence of the argument is to show there is an arbitrarily small rotation that is periodic.

Define $c_0 = 0, s_0 = 1, c_{n+1} = \sqrt{{ 1 + c_n \over 2}}, s_{n+1} = \sqrt{{ 1 - c_n \over 2}}$, $Q_n = \begin{bmatrix} c_n & - s_n \\ s_n & c_n \end{bmatrix}$. Note that $c_n, s_n \ge 0$ and $c_n^2+s_n^2 = 1$. Also, $c_n \uparrow 1$ (and hence $s_n \downarrow 0$).

($Q_0$ is a $90^\circ$ rotation, and $Q_{n+1}$ is a rotation through half of the angle of $Q_n$.)

A little work shows that $Q_n Q_n^T = I$, $Q_{n+1}^2 = Q_n$ and $Q_n \to I$. Furthermore, $Q_n^{4n} = I$ and $Q_0 Q_n^T = Q_n^T Q_0$.

Let $J= Q_0$, and $x = (g,f)^T$. Then $x' = Jx$ and $x(0) = e_1$.

Note that $x'(0) = e_2$, hence there is some $T>0$ such that $x(t) > 0$ (coordinate wise) for all $t \in (0,T]$. In particular, there is some $n$ and some $t^* \in (0,T]$ such that $x(t^*) = Q_n x(0) = Q_n e_1$.

Now consider $y(t) = Q_n^T x(t+t^*)$, note that $y(0) = x(0)$ and $y'(t) = Q_n^T J x(t+t^*) = J Q_n^T x(t+t^*)= J y(t)$ and by uniqueness we have $x(t+t^*) = Q_n x(t)$.

In particular, $x(kt^*) = Q_n^kx(0)$ and so $x(4nt) = x(0)$. Hence $x$ is periodic.