Product-to-Sum Formulas 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 $\mathbb{R}$, write $u-v$ for $u+(-v)$ as in The Real Numbers: Standing Notation and Background (i), and let $2=1+1$. Then the following hold for all $u,v\in\mathbb{R}$.

1. (Two cosines)(i)

$$\cos(u-v)+\cos(u+v)=2\cos u\cos v .$$

2. (Two sines)(ii)

$$\cos(u-v)-\cos(u+v)=2\sin u\sin v .$$

3. (A sine and a cosine)(iii)

$$\sin(u+v)+\sin(u-v)=2\sin u\cos v .$$