Trigonometric Identities

$$\sin(-\alpha) = -\sin\alpha$$ (B.26)

$$\cos(-\alpha) =+ \cos\alpha$$ (B.27)

$$\tan(-\alpha) = -\tan\alpha$$ (B.28)

$$\sin(\alpha\pm\pi/2) =\pm\cos\alpha$$ (B.29)

$$\cos(\alpha\pm\pi/2) =\mp \sin\alpha$$ (B.30)

$$\sin^2\alpha+\cos^2\alpha =1$$ (B.31)

$$\sin(\alpha\pm\beta) =\sin\alpha\,\cos\beta\pm \cos\alpha\,\sin\beta$$ (B.32)

$$\cos(\alpha\pm \beta) =\cos\alpha\,\cos\beta\mp \sin\alpha\,\sin\beta$$ (B.33)

$$\tan(\alpha\pm \beta) = \frac{\tan\alpha\pm \tan\beta}{1\mp \tan\alpha\,\tan\beta}$$ (B.34)

$$\sin\alpha+\sin\beta =2\,\sin\left(\frac{\alpha+\beta}{2}\right)\,\cos\left(\frac{\alpha-\beta}{2}\right)$$ (B.35)

$$\sin\alpha-\sin\beta =2\,\cos\left(\frac{\alpha+\beta}{2}\right)\,\sin\left(\frac{\alpha-\beta}{2}\right)$$ (B.36)

$$\cos\alpha+\cos\beta =2\,\cos\left(\frac{\alpha+\beta}{2}\right)\,\cos\left(\frac{\alpha-\beta}{2}\right)$$ (B.37)

$$\cos\alpha-\cos\beta =-2\,\sin\left(\frac{\alpha+\beta}{2}\right)\,\sin\left(\frac{\alpha-\beta}{2}\right)$$ (B.38)

$$\sin\alpha\,\sin\beta =\frac{1}{2}\left[\cos(\alpha-\beta)-\cos(\alpha+\beta)\right]$$ (B.39)

$$\cos\alpha\,\cos\beta =\frac{1}{2}\left[\cos(\alpha-\beta)+\cos(\alpha+\beta)\right]$$ (B.40)

$$\sin\alpha\,\cos\beta =\frac{1}{2}\left[\sin(\alpha-\beta)+\sin(\alpha+\beta)\right]$$ (B.41)

$$\sin(\alpha/2) = \pm\left(\frac{1-\cos\alpha}{2}\right)^{1/2}$$ (B.42)

$$\cos(\alpha/2) = \pm \left(\frac{1+\cos\alpha}{2}\right)^{1/2}$$ (B.43)

$$\tan(\alpha/2) = \pm\left(\frac{1-\cos\alpha}{1+\cos\alpha}\right)^{1/2} = \frac{1-\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1+\cos\alpha}$$ (B.44)

$$\sin(2\alpha) = 2\,\sin\alpha\,\cos\alpha$$ (B.45)

$$\cos(2\alpha) =\cos^2\alpha-\sin^2\alpha = 2\,\cos^2\alpha-1=1-2\,\sin^2\alpha$$ (B.46)

$$\sin(3\alpha) =-4\,\sin^3\alpha+3\,\sin\alpha$$ (B.47)

$$\cos(3\alpha) =4\,\cos^3\alpha-3\,\cos\alpha$$ (B.48)

$$\sin(4\alpha) = (-8\,\sin^3\alpha+4\,\sin\alpha)\,\cos\alpha$$ (B.49)

$$\cos(4\alpha) =8\,\cos^4\alpha-8\,\cos^2\alpha+1$$ (B.50)

$$\sin^2\alpha =\frac{1}{2}\,(1-\cos 2\alpha)$$ (B.51)

$$\cos^2\alpha =\frac{1}{2}\,(1+\cos 2\alpha)$$ (B.52)

$$\sin^3\alpha = \frac{1}{4}\,(3\,\sin\alpha-\sin 3\alpha)$$ (B.53)

$$\cos^3\alpha =\frac{1}{4}\,(3\,\cos\alpha+\cos 3\alpha)$$ (B.54)

$$\sin^4\alpha = \frac{1}{8}\,(3-4\,\cos 2\alpha+\cos 4\alpha)$$ (B.55)

$$\cos^4\alpha = \frac{1}{8}\,(3+4\,\cos 2\alpha+\cos 4\alpha)$$ (B.56)

$$\sinh(-\alpha) = -\sinh\alpha$$ (B.57)

$$\cosh(-\alpha) =+ \cosh\alpha$$ (B.58)

$$\tanh(-\alpha) = -\tanh\alpha$$ (B.59)

$$\cosh^2\alpha-\sinh^2\alpha =1$$ (B.60)

$$\sinh(\alpha\pm\beta) =\sinh\alpha\,\cosh\beta\pm \cosh\alpha\,\sinh\beta$$ (B.61)

$$\cosh(\alpha\pm \beta) =\cosh\alpha\,\cosh\beta\pm \sinh\alpha\,\sinh\beta$$ (B.62)

$$\tanh(\alpha\pm \beta) = \frac{\tanh\alpha\pm\tanh\beta}{1\pm\tanh\alpha\,\tanh\beta}$$ (B.63)

$$\sinh\alpha+\sinh\beta =2\,\sinh\left(\frac{\alpha+\beta}{2}\right)\,\cosh\left(\frac{\alpha-\beta}{2}\right)$$ (B.64)

$$\sinh\alpha-\sinh\beta =2\,\cosh\left(\frac{\alpha+\beta}{2}\right)\,\sinh\left(\frac{\alpha-\beta}{2}\right)$$ (B.65)

$$\cosh\alpha+\cosh\beta =2\,\cosh\left(\frac{\alpha+\beta}{2}\right)\,\cosh\left(\frac{\alpha-\beta}{2}\right)$$ (B.66)

$$\cosh\alpha-\cosh\beta =2\,\sinh\left(\frac{\alpha+\beta}{2}\right)\,\sinh\left(\frac{\alpha-\beta}{2}\right)$$ (B.67)

$$\sinh\alpha\,\sinh\beta =\frac{1}{2}\left[\cosh(\alpha+\beta)-\cosh(\alpha-\beta)\right]$$ (B.68)

$$\cosh\alpha\,\cosh\beta =\frac{1}{2}\left[\cosh(\alpha+\beta)+\cosh(\alpha-\beta)\right]$$ (B.69)

$$\sinh\alpha\,\cosh\beta =\frac{1}{2}\left[\sinh(\alpha+\beta)+\sinh(\alpha-\beta)\right]$$ (B.70)

$$\sinh(\alpha/2) =\pm \left(\frac{\cosh\alpha-1}{2}\right)^{1/2}$$ (B.71)

$$\cosh(\alpha/2) =\left(\frac{\cosh\alpha+1}{2}\right)^{1/2}$$ (B.72)

$$\tanh(\alpha/2) =\pm \left(\frac{\cosh\alpha-1}{\cosh\alpha+1}\right)^{1/2}=\frac{\cosh\alpha-1}{\sinh\alpha}= \frac{\sinh\alpha}{\cosh\alpha+1}$$ (B.73)

$$\sinh(2\alpha) = 2\,\sinh\alpha\,\cosh\alpha$$ (B.74)

$$\cosh(2\alpha) =\cosh^2\alpha+\sinh^2\alpha = 2\,\cosh^2\alpha-1=2\,\sinh^2\alpha+1$$ (B.75)

$$\cos\theta = \frac{1}{2}\left({\rm e}^{\,{\rm i}\,\theta}+{\rm e}^{-{\rm i}\,\theta}\right)$$ (B.76)

$$\sin\theta = \frac{1}{2\,{\rm i}}\left({\rm e}^{\,{\rm i}\,\theta}-{\rm e}^{-{\rm i}\,\theta}\right)$$ (B.77)

$$\cosh\theta = \frac{1}{2}\left({\rm e}^{\,\theta}+{\rm e}^{-\theta}\right)$$ (B.78)

$$\sinh\theta = \frac{1}{2}\left({\rm e}^{\,\theta}-{\rm e}^{-\theta}\right)$$ (B.79)

$$\cos({\rm i}\,\theta) = \cosh\theta$$ (B.80)

$$\sin({\rm i}\,\theta) = {\rm i}\,\sinh\theta$$ (B.81)

$$\cosh({\rm i}\,\theta) = \cos\theta$$ (B.82)

$$\sinh({\rm i}\,\theta) = {\rm i}\,\sin\theta$$ (B.83)

Source: Spiegel and Liu 1999.