# Common trigonometric identities

$$\sin ^2 \alpha +\cos ^2 \alpha = 1$$ $$\tan \alpha=\frac{\sin\alpha}{\cos\alpha}, \ \cot\alpha=\frac{\cos\alpha}{\sin\alpha}$$ $$\tan\alpha\cdot \cot\alpha=1$$
$$\sin({2\alpha})=2\sin{\alpha}\cos{\alpha}$$ $$\cos({2\alpha})=\cos^2\alpha-\sin^2\alpha$$ $$\tan({2\alpha})=\frac{2\tan\alpha}{1-\tan^2\alpha}, \ \cot({2\alpha})=\frac{\cot^2\alpha-1}{2\cot\alpha}$$
$$\sin3\alpha=-4\sin^{3}\alpha+3\sin\alpha$$ $$\cos3\alpha=4\cos^{3}\alpha-3\cos\alpha$$
$$1+\cos\alpha=2\cos^{2}\frac{\alpha}{2}$$ $$1-\cos\alpha=2\sin^{2}\frac{\alpha}{2}$$ $$1+\tan^{2}\alpha=\frac{1}{\cos^{2}\alpha}, \ 1+\cot^{2}\alpha=\frac{1}{\sin^{2}\alpha}$$
$$\tan\alpha \pm \tan\beta=\frac{\sin(\alpha \pm \beta)}{\cos\alpha\cos\beta}, \ \cot\alpha \pm \cot\beta=\frac{\sin(\beta \pm \alpha)}{\sin\alpha\sin\beta}$$ $$\sin\alpha \pm \sin\beta=2\sin\frac{\alpha \pm \beta}{2}\cos\frac{\alpha \mp \beta}{2}$$ $$\cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}$$ $$\cos\alpha-\cos\beta=2\sin\frac{\alpha+\beta}{2}\sin\frac{\beta-\alpha}{2}$$
$$\tan(\alpha \pm \beta)=\frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha \tan\beta}, \ \cot(\alpha \pm \beta)=\frac{\cot\alpha \cot\beta\mp 1}{\cot\beta \pm \tan\alpha}$$ $$\sin(\alpha \pm \beta)=\sin\alpha\cos\beta \pm \cos\alpha\sin\beta$$ $$\cos(\alpha \pm \beta)=\cos\alpha\cos\beta \mp \sin\alpha\sin\beta$$
$$\sin\alpha\cos\beta=\frac{1}{2}(\sin(\alpha-\beta)+\sin(\alpha+\beta))$$ $$\cos\alpha\cos\beta=\frac{1}{2}(\cos(\alpha-\beta)+\cos(\alpha+\beta))$$ $$\sin\alpha\sin\beta=\frac{1}{2}(\cos(\alpha-\beta)-\cos(\alpha+\beta))$$

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