Axiomatic definition of sin and cos? I look for possiblity to define sin/cos through algebraic relations without involving power series, integrals, differential equation and geometric intuition.
Is it possible to define  sin and cos through some axioms?
Like:
$$\sin 0 = 0, \cos 0 = 1$$
$$\sin \pi/2 = 1, \cos \pi/2 = 0$$
$$\sin^2 x + \cos^2 x = 1$$
$$\sin(x+2\pi n) = \sin x, \cos(x+2\pi n) = \cos x$$
$$\sin(-x)=-\sin x, \cos(-x) = \cos x \text{ for } x \in [-\pi;0]$$
$$\sin(x+y)=\sin x \cos y + \sin y \cos x$$
and be able to prove trigonometric school equations?
What additions are required to prove continuity and uniqueness of such functions and analysis properties like:
$$\lim_{x \to 0}\frac{\sin x}{x} = 0$$
or
$$\sin ' x = \cos x$$
or
$$\int \frac{dx}{\sqrt {1-x^2}} = \arcsin x$$
PS In Walter Rudin book "Principles of Mathematical analysis"  sin and cos introduced through power series.
In Solomon Feferman book "The Number Systems: Foundations of Algebra and Analysis" I see system derived from integral definition.

From what you do not want that pretty much leaves functional equations.
There seems to be a system of two functional equations for sine and cosine: (link)
$$
\Theta(x+y)=\Theta(x)\Theta(y)-\Omega(x)\Omega(y) \\
\Omega(x+y)=\Theta(x)\Omega(y)+\Omega(x)\Theta(y)
$$
