\[ \sin^2 \alpha + \cos^2 \alpha = 1 \]

\[ \mathtt{tg}^2 \alpha + 1 = \frac{1}{\cos^2 \alpha} = \mathtt{sec}^2 \alpha, \qquad \alpha \neq \frac{\pi}{2} + \pi n, n \in \mathbb Z \]

\[ \mathtt{ctg}^2 \alpha + 1 = \frac{1}{\sin^2 \alpha} = \mathtt{cosec}^2 \alpha, \qquad \alpha \neq \pi n, n \in \mathbb Z \]

\[ \mathtt{tg} \alpha \cdot \mathtt{ctg} \alpha = 1, \qquad \alpha \neq \frac{\pi n}{2}, n \in \mathbb Z \]

\[ \sin \left(\alpha \pm \beta \right) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \]

\[ \cos \left(\alpha \pm \beta \right) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta \]

\[ \mathtt{tg} \left(\alpha \pm \beta \right) = \frac{\mathtt{tg} \alpha \pm \mathtt{tg} \beta}{1 \mp \mathtt{tg} \alpha \, \mathtt{tg}\beta} \]

\[ \mathtt{ctg} \left(\alpha \pm \beta \right) = \frac{\mathtt{ctg} \alpha \, \mathtt{ctg} \beta \mp 1}{\mathtt{ctg} \beta \pm \mathtt{ctg}\alpha} \]

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