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 ${}x,y\in \mathbb {R}$ .

We have ${}\cos 0=1$ and ${}\sin 0=0$ .
We have ${}\cos {\left(-x\right)}=\cos x$ and ${}\sin {\left(-x\right)}=-\sin x$ .
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.
We have
${}(\cos x)^{2}+(\sin x)^{2}=1\,.$