Values at Zero, Parity, Addition and Double-Angle Identities for 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 $\mathbb{R}$, let $2=1+1$, and for a real number $t$ let $t^{2}=tt$ as in The Real Numbers: Standing Notation and Background (i). Then the following hold for all $x,y\in\mathbb{R}$.

1. (Values at zero)(i)

$$\cos 0=1,\qquad \sin 0=0 .$$

2. (Parity)(ii)

$$\cos(-x)=\cos x,\qquad \sin(-x)=-\sin x .$$

3. (Addition)(iii)

$$\sin(x+y)=\sin x\,\cos y+\cos x\,\sin y, \qquad \cos(x+y)=\cos x\,\cos y-\sin x\,\sin y .$$

4. (Double angle)(iv)

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

5. (A difference of squared sines)(v)

$$(\sin x)^{2}-(\sin y)^{2}=\sin(x+y)\,\sin(x-y).$$