Addition 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 $\mathbb{R}$. Then the following hold for all $a,x\in\mathbb{R}$.

1. (Cosine of a sum)(i)

$$\cos(a+x)=\cos a\cos x-\sin a\sin x .$$

2. (Sine of a sum)(ii)

$$\sin(a+x)=\sin a\cos x+\cos a\sin x .$$