Trigonometric Identities

product to sum 

    \begin{eqnarray*} \sin \alpha ~ \sin \beta &=& {1 \over 2} \left[ \cos(\alpha-\beta) - \cos(\alpha+\beta)\right] \\ \cos \alpha ~ \cos \beta &=& {1 \over 2} \left[ \cos(\alpha-\beta) + \cos(\alpha+\beta)\right] \\ \sin \alpha ~ \cos \beta &=& {1 \over 2} \left[ \sin(\alpha+\beta) + \sin(\alpha-\beta)\right] \\ \cos \alpha ~ \sin \beta &=& {1 \over 2} \left[ \sin(\alpha+\beta) - \sin(\alpha-\beta)]\right \\ \end{eqnarray*}

    \begin{eqnarray*} \sin^2\alpha = {1-\cos 2\alpha \over 2} \end{eqnarray*}

sum to product 

    \begin{eqnarray*} \sin\alpha + \sin\beta &=& {1\over2}\sin\left({\alpha+\beta \over 2}\right) \cos\left({\alpha-\beta \over 2}\right) \\ \sin\alpha - \sin\beta &=& {1\over2}\cos\left({\alpha+\beta \over 2}\right) \sin\left({\alpha-\beta \over 2}\right) \\ \cos\alpha + \cos\beta &=& {1\over2}\cos\left({\alpha+\beta \over 2}\right) \cos\left({\alpha-\beta \over 2}\right) \\ \cos\alpha - \cos\beta &=& {-1\over2}\sin\left({\alpha+\beta \over 2}\right) \sin\left({\alpha-\beta \over 2}\right) \end{eqnarray*}

    \begin{eqnarray*} \sin^2\alpha = {1-\cos 2\alpha \over 2} \end{eqnarray*}

 

sum/difference

    \begin{eqnarray*} \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) &=& {\tan\alpha \pm \tab\beta \over 1\mp \tan\alpha\tan\beta} \end{eqnarray*}

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