Trig Identities

${\sin }^2\xi +{\cos }^2\xi =1$

$\sin (\xi +\Psi )=\sin \xi \cos \Psi +\cos \xi \sin \Psi$

$\sin (\xi -\Psi )=\sin \xi \cos \Psi -\cos \xi \sin \Psi$

$\cos (\xi +\Psi )=\cos \xi \cos \Psi -\sin \xi \sin \Psi$

$\cos (\xi -\Psi )=\cos \xi \cos \Psi +\sin \xi \sin \Psi$

$2\sin \xi \sin \Psi =\cos (\xi -\Psi )-\cos (\xi +\Psi )$

$2\sin \xi \cos \Psi =\sin (\xi -\Psi )+\sin (\xi +\Psi )$

$2\cos \xi \cos \Psi =\cos (\xi -\Psi )+\cos (\xi +\Psi )$

$\sin \xi +\sin \Psi =2\sin (\frac{\xi +\Psi }{2})\cos (\frac{\xi -\Psi }{2})$

$\sin \xi -\sin \Psi =2\sin (\frac{\xi -\Psi }{2})\cos (\frac{\xi +\Psi }{2})$

$\cos \xi +\cos \Psi =2\cos (\frac{\xi +\Psi }{2})\cos (\frac{\xi -\Psi }{2})$

$\cos \xi -\cos \Psi =-2\sin (\frac{\xi -\Psi }{2})\sin (\frac{\xi +\Psi }{2})$

$\sin 2\xi =2\sin \xi \cos \xi$

$\sin (\frac{\xi }{2})=\pm \sqrt{\frac{1-\cos \xi }{2}}$

$\cos 2\xi ={\cos }^2\xi -{\sin }^2\xi =2{\cos }^2\xi -1=1-2{\sin }^2\xi$

$\cos (\frac{\xi }{2})=\pm \sqrt{\frac{1+\cos \xi }{2}}$

$\sin \xi \csc \xi =1$

$\cos \xi \sec \xi =1$

$\tan \xi \cot \xi =1$

$\tan \xi =\frac{\sin \xi }{\cos \xi }$

$\cot \xi =\frac{\cos \xi }{\sin \xi }$

${\tan }^2\xi +1={\sec }^2\xi$

${\cot }^2\xi +1={\csc }^2\xi$

$\sin (-\xi )=-\sin \xi$

$\cos (-\xi )=\cos \xi$

$\tan (-\xi )=-\tan \xi$

$\csc (-\xi )=-\csc \xi$

$\sec (-\xi )=\sec \xi$

$\cot (-\xi )=-\cot \xi$

$\tan (\xi +\Psi )=\frac{\tan \xi +\tan \Psi }{1-\tan \xi \tan \Psi }$

$\tan (\xi -\Psi )=\frac{\tan \xi -\tan \Psi }{1+\tan \xi \tan \Psi }$

$\cot (\xi +\Psi )=\frac{\cot \xi \cot \Psi -1}{\cot \xi +\cot \Psi }$

$\cot (\xi -\Psi )=-\frac{\cot \xi \cot \Psi +1}{\cot \xi -\cot \Psi }$

$\tan (2\xi )=\frac{2\tan \xi }{1-{\tan }^2\xi }$

$\cot (2\xi )=\frac{{\cot }^2\xi -1}{2\cot \xi }$

$\tan (\frac{\xi }{2})=\frac{\sin \xi }{1+\cos \xi }=\pm \sqrt{\frac{1-\cos \xi }{1+\cos \xi }}$

$\cot (\frac{\xi }{2})=\frac{\sin \xi }{1-\cos \xi }=\pm \sqrt{\frac{1+\cos \xi }{1-\cos \xi }}$