Uniform Convergence, Continuity, Parity and Derivatives of Sine and Cosine

In the setting of The Real Numbers: Standing Notation and Background, let $\cos$ and $\sin$ be the cosine and sine functions from $\mathbb{R}$ to $\mathbb{R}$, and for $k\in\mathbb{N}$ let $c_{k},\sigma_{k}:\mathbb{R}\to\mathbb{R}$ be the terms of the two series occurring there,

$$c_{k}(x)=\frac{(-1)^{k}x^{2k}}{(2k)!}, \qquad \sigma_{k}(x)=\frac{(-1)^{k}x^{2k+1}}{(2k+1)!} \qquad(x\in\mathbb{R}),$$

the exponents being the natural numbers abbreviated in Iterated Powers, Factorials, and Convergence of the Series of Powers over Factorials (i). Continuity of a function from $\mathbb{R}$ to $\mathbb{R}$, at a point or on $\mathbb{R}$, means continuity relative to $\mathbb{R}$ formed with the metric $d_{\mathbb{R}}$ of The Real Numbers: Standing Notation and Background (i); and differentiability at a point, with a given derivative, is that notion on the interval $\mathbb{R}$, every point of which is an interior point of it by Basic Facts about Intervals of the Real Line (i).

Then the following hold.

1. (Uniform convergence on a bounded interval, with a tail bound)(i) Let $R$ be a positive real number and let $[-R,R]$ be the closed interval determined by $-R$ and $R$. Then $\sum_{k=1}^{\infty}c_{k}$ converges uniformly on $[-R,R]$ to the function $x\mapsto\cos x-1$, and $\sum_{k=1}^{\infty}\sigma_{k}$ converges uniformly on $[-R,R]$ to the function $x\mapsto\sin x-x$. Moreover, for every $m\in\mathbb{N}$ and every $x\in[-R,R]$,

$$\Bigl|\cos x-1-\sum_{k=1}^{m}c_{k}(x)\Bigr|\le\sum_{k=1}^{\infty}\frac{R^{2k}}{(2k)!}-\sum_{k=1}^{m}\frac{R^{2k}}{(2k)!},$$ $$\Bigl|\sin x-x-\sum_{k=1}^{m}\sigma_{k}(x)\Bigr|\le\sum_{k=1}^{\infty}\frac{R^{2k+1}}{(2k+1)!}-\sum_{k=1}^{m}\frac{R^{2k+1}}{(2k+1)!},$$

the two infinite series converging by Iterated Powers, Factorials, and Convergence of the Series of Powers over Factorials (iv).

2. (Continuity)(ii) The functions $\cos$ and $\sin$ are continuous on $\mathbb{R}$.

3. (Values at the origin)(iii) $\cos 0=1$ and $\sin 0=0$.

4. (Parity)(iv) For every $x\in\mathbb{R}$ one has $\cos(-x)=\cos x$ and $\sin(-x)=-\sin x$.

5. (Derivatives)(v) For every $x\in\mathbb{R}$ the function $\sin$ is differentiable at $x$ with derivative $\cos x$, and the function $\cos$ is differentiable at $x$ with derivative $-\sin x$.