Sine and cosine/Real/Properties/Fact

The functions

$ℝ⟶ℝ,x⟼\cos x,$

and

$ℝ⟶ℝ,x⟼\sin x,$

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

1. We have $\cos 0=1$ and $\sin 0=0$.
2. We have $\cos (-x)=\cos x$ and $\sin (-x)=-\sin x$.
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