# Signals and Systems/Table of Trigonometric Identities

## Useful Mathematical Identities

 ${\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1}$ ${\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta }$ ${\displaystyle \sin({\frac {\pi }{2}}-\theta )=\cos \theta }$ ${\displaystyle \cos({\frac {\pi }{2}}-\theta )=\sin \theta }$ ${\displaystyle \sec({\frac {\pi }{2}}-\theta )=\csc \theta }$ ${\displaystyle \csc({\frac {\pi }{2}}-\theta )=\sec \theta }$ ${\displaystyle \sin(-\theta )=-\sin \theta }$ ${\displaystyle \cos(-\theta )=\cos \theta }$ ${\displaystyle \sin 2\theta =2\sin \theta \cos \theta }$ ${\displaystyle \cos 2\theta =\cos ^{2}-\sin ^{2}=2\cos ^{2}\theta -1=1-2\sin ^{2}\theta }$ ${\displaystyle \sin ^{2}\theta ={\frac {1-\cos 2\theta }{2}}}$ ${\displaystyle \cos ^{2}\theta ={\frac {1+\cos 2\theta }{2}}}$ ${\displaystyle \sin \alpha +\sin \beta =2\sin({\frac {\alpha +\beta }{2}})\cos({\frac {\alpha -\beta }{2}})}$ ${\displaystyle \sin \alpha -\sin \beta =2\cos({\frac {\alpha +\beta }{2}})\sin({\frac {\alpha -\beta }{2}})}$ ${\displaystyle \cos \alpha +\cos \beta =2\cos({\frac {\alpha +\beta }{2}})\cos({\frac {\alpha -\beta }{2}})}$ ${\displaystyle \cos \alpha -\cos \beta =-2\sin({\frac {\alpha +\beta }{2}})\sin({\frac {\alpha -\beta }{2}})}$ ${\displaystyle \sin \alpha \sin \beta ={\frac {1}{2}}[\cos(\alpha -\beta )-\cos(\alpha +\beta )]}$ ${\displaystyle \cos \alpha \cos \beta ={\frac {1}{2}}[\cos(\alpha -\beta )+\cos(\alpha +\beta )]}$ ${\displaystyle \sin \alpha \cos \beta ={\frac {1}{2}}[\sin(\alpha +\beta )+\sin(\alpha -\beta )]}$ ${\displaystyle 1+\cot ^{2}=\csc ^{2}}$ ${\displaystyle e^{j\theta }=\cos \theta +j\sin \theta }$ ${\displaystyle \cos \theta ={\frac {e^{j\theta }+e^{-j\theta }}{2}}}$ ${\displaystyle e^{-j\theta }=\cos \theta -j\sin \theta }$ ${\displaystyle \sin \theta ={\frac {e^{j\theta }-e^{-j\theta }}{2j}}}$ ${\displaystyle \tan({\frac {\pi }{2}}-\theta )=\cot \theta }$ ${\displaystyle \cot({\frac {\pi }{2}}-\theta )=\tan \theta }$ ${\displaystyle \tan(-\theta )=-\tan \theta }$ ${\displaystyle \tan ^{2}\theta ={\frac {1-\cos 2\theta }{1+\cos 2\theta }}}$ ${\displaystyle \tan 2\theta ={\frac {2\tan \theta }{1-tan^{2}\theta }}}$