The sum (difference) of sines \[ \sin \alpha \pm \sin \beta = 2 \sin \frac{\alpha \pm \beta}{2} \cos \frac{\alpha \mp \beta}{2} \]

The sum of the cosines \[ \cos \alpha + \cos \beta = 2 \cos \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2} \]

The difference of the cosines \[ \cos \alpha - \cos \beta = - 2 \sin \frac{\alpha + \beta}{2} \sin \frac{\alpha - \beta}{2} \]

The sum of the tangents \[ \mathtt{tg} \alpha \pm \mathtt{tg} \beta = \frac{\sin (\alpha \pm \beta)}{\cos \alpha \cos \beta} \]

The difference between the cotangent \[ \mathtt{ctg} \alpha \pm \mathtt{ctg} \beta = \frac{\sin (\beta \pm \alpha)}{\sin \alpha \sin \beta} \]

The product or sum of three sines \[ \sin 2\alpha + \sin 2\beta + \sin 2\gamma = 4 \sin\alpha \sin\beta \sin\gamma \]

The sine double angle \[ \sin 2 \alpha = 2 \sin\alpha \, \cos\alpha \]

The cosine double angle \[ \cos 2 \alpha = \cos^2 \alpha - \sin^2 \alpha = 2 \cos^2 \alpha - 1 = 1 - 2 \sin^2 \alpha \]

Tangent double angle \[ \mathtt{tg} 2 \alpha = \frac{2\,\mathtt{tg} \alpha}{1 - \mathtt{tg}^2 \alpha} \]

Cotangent double angle \[ \mathtt{ctg} 2 \alpha = \frac{\mathtt{ctg}^2 \alpha - 1}{2\,\mathtt{ctg} \alpha} \]

The sine of a triple angle \[ \sin 3\alpha = 3 \sin \alpha - 4 \sin^3\alpha \]

The cosine triple angle \[ \cos 3\alpha = 4 \cos^3\alpha - 3 \cos \alpha \]

The triple tangent of the angle \[ \mathtt{tg} 3\alpha = \frac{3\,\mathtt{tg}\alpha - \mathtt{tg}^3\alpha}{1 - 3\,\mathtt{tg}^2\alpha} \]

The cotangent triple angle \[ \mathtt{ctg} 3\alpha = \frac{3\,\mathtt{ctg}\alpha - \mathtt{ctg}^3\alpha}{1 - 3\,\mathtt{ctg}^2\alpha} \]

The sine of the half angle \[ \sin {\biggl (}{\alpha \over 2}{\biggr )} = \pm {\sqrt{1-\cos \alpha \over 2}} \]

The cosine of the half angle \[ \cos {\biggl (}{\alpha \over 2}{\biggr )} = \pm {\sqrt{1+\cos \alpha \over 2}} \]

The tangent of the half angle \[ \mathtt{tg}{\biggl (}{\alpha \over 2}{\biggr)} = \pm {\sqrt{1-\cos \alpha \over 1+\cos \alpha}} = {\sin \alpha \over 1+\cos \alpha} = {1-\cos \alpha \over \sin \alpha} \]