Сумма и разность тангенсов

\[\mathrm{tg}{\alpha} + \mathrm{tg}{\beta} = \frac{\sin(\alpha+ \beta)}{\cos{\alpha} \cos{\beta}}\]

\[\mathrm{tg}{\alpha} - \mathrm{tg}{\beta} = \frac{\sin(\alpha- \beta)}{\cos{\alpha} \cos{\beta}}\]

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

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

\[\cos{\alpha} + \cos{\beta} = 2 \cos{\frac{\alpha + \beta}{2}} \cos{\frac{\alpha - \beta}{2}}\]

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

\[\mathrm{ctg}{\alpha} + \mathrm{ctg}{\beta} = \frac{\sin(\alpha + \beta)}{\sin{\alpha} \cdot \sin{\beta}}\]

\[\mathrm{ctg}{\alpha} - \mathrm{ctg}{\beta} = \frac{-\sin(\alpha - \beta)}{\sin{\alpha} \cdot \sin{\beta}}\]