Characterisation of Sine and Cosine

Theorem
Let $\map s x: \R \to \R$, $\map c x: \R \to \R$ be differentiable real functions that satisfy:


 * $(1): \quad \map {s'} x = \map c x$
 * $(2): \quad \map {c'} x = -\map s x$
 * $(3): \quad \map s 0 = 0$
 * $(4): \quad \map c 0 = 1$
 * $(5): \quad \forall x: \map {s^2} x + \map {c^2} x = 1$

Then, for every $x \in \R$:
 * $\map s x = \map \sin x$
 * $\map c x = \map \cos x$

Proof
Define:
 * $\map h x = \paren {\map c x - \map \cos x}^2 + \paren {\map s x - \map \sin x}^2$

Then:

By taking $\map {h'} x$:

By Zero Derivative implies Constant Function, $\map h x$ is a constant function:
 * $\map h x = k$

Also:

Since $\map h x$ is constant:
 * $\map h x = 0$

Therefore:
 * $\paren {\map c x - \map \cos x}^2 + \paren {\map s x - \map \sin x}^2 = 0$

for every $x \in \R$.

But since Square of Real Number is Non-Negative and Square of Non-Zero Real Number is Strictly Positive:
 * $\map c x - \map \cos x = 0$
 * $\map s x - \map \sin x = 0$

The result follows.