Characterisation of Sine and Cosine

Theorem
The definitions for sine and cosine are equivalent.

That is:


 * $\displaystyle \sin x = \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac {x^{2 n + 1} } {\left({2 n + 1}\right)!} \iff \sin x = \frac {\text{Opposite}} {\text{Hypotenuse}}$


 * $\displaystyle \cos x = \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac {x^{2 n} } {\left({2 n}\right)!} \iff \cos x = \frac {\text{Adjacent}} {\text{Hypotenuse}}$

Proof
Let $s \left({x}\right): \R \to \R$, $c \left({x}\right): \R \to \R$ be two functions that satisfy:


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

where $s'$ denotes the derivative $x$.

Let $f \left({x}\right): \R \to \R$, $g \left({x}\right): \R \to \R$ also be two functions that satisfy:
 * $(1): \quad f' \left({x}\right) = g \left({x}\right)$
 * $(2): \quad g' \left({x}\right) = -f \left({x}\right)$
 * $(3): \quad f \left({0}\right) = 0$
 * $(4): \quad g \left({0}\right) = 1$
 * $(5): \quad \forall x: f^2 \left({x}\right) + g^2 \left({x}\right) = 1$

It will be shown that:
 * $f \left({x}\right) = s \left({x}\right)$

and:
 * $c \left({x}\right) = g \left({x}\right)$

Define:
 * $h \left({x}\right) = \left({c \left({x}\right) - g \left({x}\right)}\right)^2 + \left({s \left({x}\right) - f \left({x}\right)}\right)^2$

Notice that:
 * $\left({\forall x: h \left({x}\right) = 0}\right) \iff \left({\forall x: c \left({x}\right) = g \left({x}\right), s \left({x}\right) = f \left({x}\right)}\right)$

Then:

By taking $h' \left({x}\right)$:

So $h \left({x}\right)$ is a constant function:
 * $h \left({x}\right) = k$

Also:

Since $h \left({x}\right)$ is constant, then:
 * $\forall x: h \left({x}\right) = 0$

Then:
 * $c \left({x}\right) = g \left({x}\right)$

and:
 * $s \left({x}\right) = f \left({x}\right)$

By:
 * Derivative of Sine Function
 * Derivative of Cosine Function
 * Sine of Zero is Zero
 * Cosine of Zero is One
 * Sum of Squares of Sine and Cosine

both definitions satisfy all these properties.

Therefore they must be the same.