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,

the exponents being the natural numbers abbreviated in Iterated Powers, Factorials, and Convergence of the Series of Powers over Factorials §powers. 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 §numbers; 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 and Their Interior Points §whole-line.

Then the following hold.

1. (Uniform convergence on a bounded interval, with a tail bound)¶ 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 $x\in[-R,R]$,

the two infinite series converging by Iterated Powers, Factorials, and Convergence of the Series of Powers over Factorials §trigonometric.

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

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

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

5. (Derivatives)¶ 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 $-\sin x$.