Sine and cosine/Real/Properties/Fact

The functions

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \cos x,$

and

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \sin x,$

have the following properties for ${}x,y\in \mathbb {R}$.

1. We have ${}\cos 0=1$ and ${}\sin 0=0$.
2. We have ${}\cos {\left(-x\right)}=\cos x$ and ${}\sin {\left(-x\right)}=-\sin x$.
3. The addition theorems

   ${}\cos(x+y)=\cos x\cdot \cos y-\sin x\cdot \sin y\,$

   and

   ${}\sin(x+y)=\sin x\cdot \cos y+\cos x\cdot \sin y\,$

   hold.

4. We have

   ${}(\cos x)^{2}+(\sin x)^{2}=1\,.$

Proof, Write another proof