Step By Step Calculus » 5.4 - Addition and Subtraction Formulae

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Synopsis
Sin/Cos Conversion Formulae
\displaystyle \cos \left(\theta \right)=\sin \left(\theta +\frac{\pi }{2}\right)\displaystyle \cos \left(\theta \right)=\sin \left(\theta +\frac{\pi }{2}\right) and \displaystyle \sin \left(\theta \right)=\cos \left(\theta -\frac{\pi }{2}\right)\displaystyle \sin \left(\theta \right)=\cos \left(\theta -\frac{\pi }{2}\right)
Addition Formulae The addition formulae are as follows:
\displaystyle \begin{array}{l} \cos(\alpha + \beta) = \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta) \\ \sin(\alpha + \beta) = \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta) \\ \tan(\alpha+\beta)=\frac{\tan(\alpha)+\tan(\beta)}{1-\tan(\alpha)\tan(\beta)} \end{array}
\displaystyle \begin{array}{l} \cos(\alpha + \beta) = \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta) \\ \sin(\alpha + \beta) = \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta) \\ \tan(\alpha+\beta)=\frac{\tan(\alpha)+\tan(\beta)}{1-\tan(\alpha)\tan(\beta)} \end{array}
Subtraction Formulae The subtraction formulae are as follows:
\displaystyle \begin{array}{l} \cos(\alpha - \beta) = \cos(\alpha) \cos(\beta) + \sin(\alpha) \sin(\beta) \\ \sin(\alpha - \beta) = \sin(\alpha) \cos(\beta) - \cos(\alpha) \sin(\beta) \\ \tan(\alpha-\beta)=\frac{\tan(\alpha)-\tan(\beta)}{1+\tan(\alpha)\tan(\beta)} \end{array}
\displaystyle \begin{array}{l} \cos(\alpha - \beta) = \cos(\alpha) \cos(\beta) + \sin(\alpha) \sin(\beta) \\ \sin(\alpha - \beta) = \sin(\alpha) \cos(\beta) - \cos(\alpha) \sin(\beta) \\ \tan(\alpha-\beta)=\frac{\tan(\alpha)-\tan(\beta)}{1+\tan(\alpha)\tan(\beta)} \end{array}