\documentclass[12pt,a4paper]{letter}
\begin{document}

$\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$ \\
$\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$ \\
$\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}$ \\

$\sin\alpha + \sin\beta = 2\sin\frac{\alpha + \beta}{2}\cos\frac{\alpha - \beta}{2}$ \\
$\sin\alpha - \sin\beta = 2\cos\frac{\alpha + \beta}{2}\sin\frac{\alpha - \beta}{2}$ \\
$\cos\alpha + \cos\beta = 2\cos\frac{\alpha + \beta}{2}\cos\frac{\alpha - \beta}{2}$ \\
$\cos\alpha - \cos\beta = -2\sin\frac{\alpha + \beta}{2}\sin\frac{\alpha - \beta}{2}$ \\

$2\sin\alpha\cos\beta = \sin(\alpha + \beta) + \sin(\alpha - \beta)$ \\
$2\cos\alpha\sin\beta = \sin(\alpha + \beta) - \sin(\alpha - \beta)$ \\
$2\cos\alpha\cos\beta = \cos(\alpha + \beta) + \cos(\alpha - \beta)$ \\
$-2\sin\alpha\sin\beta = \cos(\alpha + \beta) - \cos(\alpha - \beta)$ \\

$\sin\frac\alpha2 = \pm\sqrt\frac{1 - \cos\alpha}2$ \\
$\cos\frac\alpha2 = \pm\sqrt\frac{1 + \cos\alpha}2$ \\
$\tan\frac\alpha2 = \pm\sqrt\frac{1 - \cos\alpha}{1 + \cos\alpha} = \frac{1 - \cos\alpha}{\sin\alpha} = \frac{\sin\alpha}{1 + \cos\alpha}$ \\

$\sin2\alpha = 2\sin\alpha\cos\alpha$ \\
$\cos2\alpha = 2\cos^2\alpha - 1 = 1 - 2\sin^2\alpha$ \\
$\tan2\alpha = \frac{2\tan\alpha}{1 - \tan^2\alpha}$ \\

$\sin3\alpha = 3\sin\alpha - 4\sin^3\alpha$ \\
$\cos3\alpha = 4\cos^3\alpha - 3\cos\alpha$ \\
$\tan3\alpha = \frac{3\tan\alpha - \tan^3\alpha}{1 - 3\tan^2\alpha}$ \\

\end{document}