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［加法公式］
\[
\sin \left( {\alpha \pm \beta } \right) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta
\]
\[
\cos \left( {\alpha \pm \beta } \right) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta
\]
\[
\tan \left( {\alpha \pm \beta } \right) = \frac{{\tan \alpha \pm \tan \beta }}{{1 \mp \tan \alpha \tan \beta }}
\]
\[
\cot \left( {\alpha \pm \beta } \right) = \frac{{\cot \alpha \cot \beta \mp 1}}{{\cot \beta \pm \cot \alpha }}
\]
\[
\begin{array}{l}
\sin \left( {\alpha + \beta + \gamma } \right) = \sin \alpha \cos \beta \cos \gamma + \cos \alpha \sin \beta \cos \gamma \\
+ \cos \alpha \cos \beta \sin \gamma - \sin \alpha \sin \beta \sin \gamma \\
= \sin \alpha \sin \beta \sin \gamma \left( {\cot \beta \cot \gamma + \cot \gamma \cot \alpha + \cot \alpha \cot \beta - 1} \right) \\
= \cos \alpha \cos \beta \cos \gamma \left( {\tan \alpha + \tan \beta + \tan \gamma - \tan \alpha \tan \beta \tan \gamma } \right) \\
\end{array}
\]
\[
\begin{array}{l}
\cos \left( {\alpha + \beta + \gamma } \right) = \cos \alpha \cos \beta \cos \gamma - \cos \alpha \sin \beta \sin \gamma \\
- \sin \alpha \cos \beta \sin \gamma - \sin \alpha \sin \beta \cos \gamma \\
= \cos \alpha \cos \beta \cos \gamma \left( {1 - \tan \beta \tan \gamma - \tan \gamma \tan \alpha - \tan \alpha \tan \beta } \right) \\
= - \sin \alpha \sin \beta \sin \gamma \left( {\cot \alpha + \cot \beta + \cot \gamma - \cot \alpha \cot \beta \cot \gamma } \right) \\
\end{array}
\]
［和差与积互化公式］
\[
\sin \alpha + \sin \beta = 2\sin \frac{{\alpha + \beta }}{2}\cos \frac{{\alpha - \beta }}{2}
\]
\[
\sin \alpha - \sin \beta = 2\cos \frac{{\alpha + \beta }}{2}\sin \frac{{\alpha - \beta }}{2}
\]
\[
\cos \alpha + \cos \beta = 2\cos \frac{{\alpha + \beta }}{2}\cos \frac{{\alpha - \beta }}{2}
\]
\[
\cos \alpha - \cos \beta = - 2\sin \frac{{\alpha + \beta }}{2}\sin \frac{{\alpha - \beta }}{2}
\]
\[
\tan \alpha \pm \tan \beta = \frac{{\sin \left( {\alpha \pm \beta } \right)}}{{\cos \alpha \cos \beta }}
\]
\[
\cot \alpha \pm \cot \beta = \pm \frac{{\sin \left( {\alpha \pm \beta } \right)}}{{\sin \alpha \sin \beta }}
\]
\[
\tan \alpha \pm \cot \beta = \pm \frac{{\cos \left( {\alpha \mp \beta } \right)}}{{\cos \alpha \sin \beta }}
\]
\[
\sin \alpha \sin \beta = - \frac{1}{2}\left[ {\cos \left( {\alpha + \beta } \right) - \cos \left( {\alpha - \beta } \right)} \right]
\]
\[
\cos \alpha \cos \beta = \frac{1}{2}\left[ {\cos \left( {\alpha + \beta } \right) + \cos \left( {\alpha - \beta } \right)} \right]
\]
\[
\sin \alpha \cos \beta = \frac{1}{2}\left[ {\sin \left( {\alpha + \beta } \right) + \sin \left( {\alpha - \beta } \right)} \right]
\]
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