Sine and cosine/R/Properties of periodicity/Fact

The sine function and the cosine function fulfil in ${}\mathbb {R}$ the following periodicity properties.

1. We have ${}\cos {\left(x+2\pi \right)}=\cos x$ and ${}\sin {\left(x+2\pi \right)}=\sin x$ for all ${}x\in \mathbb {R}$.
2. We have ${}\cos {\left(x+\pi \right)}=-\cos x$ and ${}\sin {\left(x+\pi \right)}=-\sin x$ for all ${}x\in \mathbb {R}$.
3. We have ${}\cos {\left(x+\pi /2\right)}=-\sin x$ and ${}\sin {\left(x+\pi /2\right)}=\cos x$ for all ${}x\in \mathbb {R}$.
4. We have ${}\cos 0=1$, ${}\cos \pi /2=0$, ${}\cos \pi =-1$, ${}\cos 3\pi /2=0$ and ${}\cos 2\pi =1$.
5. We have ${}\sin 0=0$, ${}\sin \pi /2=1$, ${}\sin \pi =0$, ${}\sin 3\pi /2=-1$ and ${}\sin 2\pi =0$.

Proof, Write another proof