$$\begin{aligned} \sin(A+B) &= \sin A\cos B + \cos A\sin B \\ \sin(A-B) &= \sin A\cos B - \cos A\sin B \\ \cos(A+B) &= \cos A\cos B - \sin A\sin B \\ \cos(A-B) &= \cos A\cos B + \sin A\sin B \\ \tan(A+B) &= \frac{\tan A + \tan B}{1 - \tan A \tan B} \\ \tan(A-B) &= \frac{\tan A - \tan B}{1 + \tan A \tan B} \\ 1 + \tan^2 x &= \sec^2 x \\ \end{aligned}$$

Note for $\sin(A+B)$ and $\cos(A+B)$, Khan Academy has some nice proofs. Notice $\cos -\theta = \cos \theta$ and $\sin -\theta = - \sin \theta$. The other identities are not hard to verify.

Euler's Formula

On the other hand, you may never have to memorize these formulas. Here is why. By Euler's formula:

$$\begin{aligned} e^{i(x+y)} &= \cos(x+y) + i \sin(x+y) \\ &= e^{ix} e^{iy} = (\cos x + i \sin x) \cdot (\cos y + i \sin y) \\ &= (\cos x \cdot \cos y - \sin x \cdot \sin y) + i(\sin x \cdot \cos y + \cos x \cdot \sin y) \\ \end{aligned}$$

and therefore

$$\begin{aligned} \cos(x+y) &= \cos x \cdot \cos y - \sin x \cdot \sin y \\ \sin(x+y) &= \sin x \cdot \cos y + \cos x \cdot \sin y \\ \end{aligned}$$