Rigorous proof of the Taylor expansions of sin $x$ and cos $x$ We learn trigonometric functions in high school, but their treatment is not rigorous.
Then we learn that they can be defined by power series in a college.
I think there is a gap between the two.
I'd like to fill in the gap in the following way.
Consider the upper right quarter of the unit circle $C$ = {$(x, y) \in \mathbb{R}^2$; $x^2 + y^2 = 1$, $0 \leq x \leq 1$, $y \geq 0$}.
Let $\theta$ be the arc length of $C$ from $x = 0$ to $x$, where $0 \leq x \leq 1$.
By the arc length formula, $\theta$ can be expressed by a definite integral of a simple algebraic function from 0 to $x$.
Clearly $\sin \theta = x$ and $\cos\theta = \sqrt{1 - \sin^2 \theta}$.
Then how do we prove that the Taylor expansions of $\sin\theta$ and $\cos\theta$ are the usual ones?

there is a simplified elementary derivation of the power series without the use of Taylor Series. It can be done through the expansion of the multiple angle formula. See paper by  David Bhatt,  “Elementary Derivation of Sine and Cosine Series”, Bulletin of the Marathwada Mathematical Society, 9(2)  2008, 10–12
