Trigonometric Functions

sin⁡2α+cos⁡2α=1\sin^2 \alpha + \cos^2 \alpha = 1
sin⁡(α±β)=sin⁡αcos⁡β±cos⁡αsin⁡β\sin (\alpha \pm \beta) = \sin\alpha \cos\beta \pm \cos\alpha\sin\beta
cos⁡(α±β)=cos⁡αcos⁡β∓sin⁡αsin⁡β\cos (\alpha \pm \beta) = \cos\alpha \cos\beta \mp \sin\alpha\sin\beta
tan⁡(α±β)=tan⁡α±tan⁡β1∓tan⁡αtan⁡β\tan (\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1\mp \tan\alpha\tan\beta}
sin⁡(2α)=2sin⁡αcos⁡α\sin (2\alpha) = 2\sin\alpha\cos\alpha
cos⁡(2α)=cos⁡2α−sin⁡2α=2cos⁡2α−1=1−2sin⁡2α\cos (2\alpha) = \cos^2\alpha-\sin^2\alpha = 2\cos^2\alpha -1 = 1 - 2\sin^2\alpha
sin⁡(3α)=3sin⁡α−4sin⁡3α\sin (3\alpha) = 3\sin\alpha - 4\sin^3\alpha
cos⁡(3α)=4cos⁡3α−3cos⁡α\cos (3\alpha) = 4\cos^3\alpha - 3\cos\alpha
sin⁡2α2=12(1−cos⁡α)\sin^2 \frac{\alpha}{2} = \frac{1}{2}(1-\cos\alpha)
cos⁡2α2=12(1+cos⁡α)\cos^2 \frac{\alpha}{2} = \frac{1}{2}(1+\cos\alpha)
sin⁡α+cos⁡β=2sin⁡12(α+β)cos⁡12(α−β)\sin\alpha + \cos\beta = 2\sin\frac{1}{2}(\alpha+\beta)\cos\frac{1}{2}(\alpha-\beta)
sin⁡α−cos⁡β=2cos⁡12(α+β)sin⁡12(α−β)\sin\alpha - \cos\beta = 2\cos\frac{1}{2}(\alpha+\beta)\sin\frac{1}{2}(\alpha-\beta)
cos⁡α+cos⁡β=2cos⁡12(α+β)cos⁡12(α−β)\cos\alpha + \cos\beta = 2\cos\frac{1}{2}(\alpha+\beta)\cos\frac{1}{2}(\alpha-\beta)
cos⁡α−cos⁡β=−2sin⁡12(α+β)sin⁡12(α−β)\cos\alpha - \cos\beta = -2\sin\frac{1}{2}(\alpha+\beta)\sin\frac{1}{2}(\alpha-\beta)
sin⁡α=12i(eiα−e−iα)\sin\alpha = \frac{1}{2i} (e^{i\alpha}-e^{-i\alpha})
cos⁡α=12(eiα+e−iα)\cos\alpha = \frac{1}{2} (e^{i\alpha}+e^{-i\alpha})