Trigonometric function

1. Basic formula

   $\begin{array}{l}{\sin }^2\alpha +{\cos }^2\alpha =1\\ \tan \alpha *\mathrm{cottan}\alpha =1\\ \tan \alpha =\frac{\sin \alpha }{\cos \alpha }=\frac{1}{\mathrm{cottan}\alpha }\\ \mathrm{cottan}\alpha =\frac{\cos \alpha }{\sin \alpha }=\frac{1}{\tan \alpha }\\ 1+{\tan }^2\alpha =\frac{1}{{\cos }^2\alpha }={\sec }^2\alpha \\ 1+{\mathrm{cottan}}^2\alpha =\frac{1}{{\sin }^2\alpha }={\mathrm{cossec}}^2\alpha \end{array}$

2. Addition formula

   $\begin{array}{l}\sin (\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\ \cos (\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta \\ \tan (\alpha \pm \beta )=\frac{\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }\end{array}$

3. The sum of trigonometric function

   $\begin{array}{l}\cos \alpha +\cos \beta =2\cos \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}\\ \cos \alpha -\cos \beta =-2\sin \frac{\alpha +\beta }{2}\sin \frac{\alpha -\beta }{2}\\ \sin \alpha +\sin \beta =2\sin \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}\\ \sin \alpha -\sin \beta =2\cos \frac{\alpha +\beta }{2}\sin \frac{\alpha -\beta }{2}\\ \sin \alpha +\cos \alpha =\sqrt{2}\sin (\alpha +\frac{\pi }{4})=\sqrt{2}\cos (\frac{\pi }{4}-\alpha )\\ \sin \alpha -\cos \alpha =\sqrt{2}\sin (\alpha -\frac{\pi }{4})=-\sqrt{2}\cos (\frac{\pi }{4}-\alpha )\\ \tan \alpha +\tan \beta =\frac{\sin (\alpha +\beta )}{\cos \alpha \cos \beta }\\ \tan \alpha -\tan \beta =\frac{\sin (\alpha -\beta )}{\cos \alpha \cos \beta }\\ \cot \tan \alpha +\cot \tan \beta =\frac{\sin (\alpha -\beta )}{\sin \alpha \sin \beta }\\ \cot \tan \alpha -\cot \tan \beta =\frac{\sin (\beta -\alpha )}{\sin \alpha \sin \beta }\\ \tan \alpha +\cot \tan \alpha =2\cos \sec 2\alpha \\ \tan \alpha -\cot \tan \alpha =-2\cos \sec 2\alpha \end{array}$

4. The product of trigonometric function

   $\begin{array}{l}\cos \alpha \cos \beta =\frac{1}{2}[\cos (\alpha -\beta )+\cos (\alpha +\beta )]\\ \sin \alpha \sin \beta =\frac{1}{2}[\cos (\alpha -\beta )-\cos (\alpha +\beta )]\\ \sin \alpha \cos \beta =\frac{1}{2}[\sin (\alpha +\beta )+\sin (\alpha -\beta )]\\ \tan \alpha \tan \beta =\frac{\tan \alpha +\tan \beta }{\cot \tan \alpha +\cot \tan \beta }=-\frac{\tan \alpha -\tan \beta }{\cot \tan \alpha +\cot \tan \beta }\\ \cot \tan \alpha \cot \tan \beta =\frac{\cot \tan \alpha +\cot \tan \beta }{\tan \alpha +\tan \beta }=-\frac{\cot \tan \alpha -\cot \tan \beta }{\tan \alpha -\tan \beta }\\ \cot \tan \alpha \tan \beta =\frac{\cot \tan \alpha +\tan \beta }{\tan \alpha +\cot \tan \beta }=-\frac{\cot \tan \alpha -\tan \beta }{\tan \alpha -\cot \tan \beta }\end{array}$

5. The power of trigonometric function

   $\begin{array}{l}{\sin }^2\alpha =\frac{1}{2}(1-\cos 2\alpha )\\ {\cos }^2\alpha =\frac{1}{2}(1+\cos 2\alpha )\\ {\tan }^2\alpha =\frac{1-\cos 2\alpha }{1+\cos 2\alpha }\\ {\sin }^3\alpha =\frac{1}{4}(3\sin \alpha -\sin 3\alpha )\\ {\cos }^3\alpha =\frac{1}{4}(3\cos \alpha +\cos 3\alpha )\end{array}$

6. Double angle formula

   $\begin{array}{l}\sin 2\alpha =2\sin \alpha \cos \alpha \\ \cos 2\alpha =2{\cos }^2\alpha -1=1-2{\sin }^2\alpha ={\cos }^2\alpha -{\sin }^2\alpha \\ \tan 2\alpha =\frac{2\tan \alpha }{1-{\tan }^2\alpha }\\ \cot \tan 2\alpha =\frac{\cot {\tan }^2\alpha -1}{2\cot \tan \alpha }=\frac{\cot \tan \alpha -\tan \alpha }{2}\\ \sin 3\alpha =3\sin \alpha -4{\sin }^3\alpha \\ \cos 3\alpha =4{\cos }^3\alpha -3\cos \alpha \\ \tan 3\alpha =\frac{3\tan \alpha -{\tan }^3\alpha }{1-3{\tan }^2\alpha }\\ \cot \tan 3\alpha =\frac{\cot {\tan }^3\alpha -3\cot \tan \alpha }{3\cot {\tan }^2\alpha -1}\end{array}$

7. Half angle formula

   $\begin{array}{l}\sin \frac{\alpha }{2}=\pm \sqrt{\frac{1-\cos \alpha }{2}}\\ \cos \frac{\alpha }{2}=\pm \sqrt{\frac{1+\cos \alpha }{2}}\\ \tan \frac{\alpha }{2}=\frac{\sin \alpha }{1+\cos \alpha }=\frac{1-\cos \alpha }{\sin \alpha }=\pm \sqrt{\frac{1-\cos \alpha }{1+\cos \alpha }}\\ \cot \tan \frac{\alpha }{2}=\frac{\sin \alpha }{1-\cos \alpha }=\frac{1+\cos \alpha }{\sin \alpha }=\pm \sqrt{\frac{1+\cos \alpha }{1-\cos \alpha }}\\ \sin \alpha =\frac{2\tan \frac{\alpha }{2}}{1+{\tan }^2\frac{\alpha }{2}}\\ \cos \alpha =\frac{1-{\tan }^2\frac{\alpha }{2}}{1+{\tan }^2\frac{\alpha }{2}}\\ \tan \alpha =\frac{2\tan \frac{\alpha }{2}}{1-{\tan }^2\frac{\alpha }{2}}\\ |\cos \alpha \pm \sin \alpha |=\sqrt{1+\sin 2\alpha }\\ 1+\cos \alpha =2{\cos }^2\frac{\alpha }{2}\\ 1-\cos \alpha =2{\sin }^2\frac{\alpha }{2}\\ 1+\sin \alpha ={(\sin \frac{\alpha }{2}+\cos \frac{\alpha }{2})}^2=2{\cos }^2(\frac{\pi }{4}-\frac{\alpha }{2})\\ 1-\sin \alpha ={(\sin \frac{\alpha }{2}-\cos \frac{\alpha }{2})}^2=2{\sin }^2(\frac{\pi }{4}-\frac{\alpha }{2})\end{array}$

8. Interior angle formula

   $\begin{array}{l}\alpha +\beta +\gamma =180°\\ \sin \alpha +\sin \beta +\sin \gamma =4\cos \frac{\alpha }{2}\cos \frac{\beta }{2}\cos \frac{\gamma }{2}\\ \cos \alpha +\cos \beta +\cos \gamma =4\sin \frac{\alpha }{2}\sin \frac{\beta }{2}\sin \frac{\gamma }{2}+1\\ \sin \alpha +\sin \beta -\cos \gamma =4\sin \frac{\alpha }{2}\sin \frac{\beta }{2}\cos \frac{\gamma }{2}\\ \cos \alpha +\cos \beta -\cos \gamma =4\cos \frac{\alpha }{2}\cos \frac{\beta }{2}\sin \frac{\gamma }{2}-1\\ {\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma =2\cos \alpha \cos \beta \cos \gamma +2\\ {\sin }^2\alpha +{\sin }^2\beta -{\sin }^2\gamma =2\sin \alpha \sin \beta \sin \gamma \\ \sin 2\alpha +\sin 2\beta +\sin 2\gamma =4\sin \alpha \sin \beta \sin \gamma \\ \sin 2\alpha +\sin 2\beta -\sin 2\gamma =4\cos \alpha \cos \beta \sin \gamma \\ \tan \alpha +\tan \beta +\tan \gamma =\tan \alpha \tan \beta \tan \gamma \\ \cot \tan \frac{\alpha }{2}+\cot \tan \frac{\beta }{2}+\cot \tan \frac{\gamma }{2}=\cot \tan \frac{\alpha }{2}\cot \tan \frac{\beta }{2}\cot \tan \frac{\gamma }{2}\\ \cot \tan \alpha \cot \tan \beta +\cot \tan \beta \cot \tan \gamma +\cot \tan \alpha \cot \tan \gamma =1\end{array}$