# Engineering Tables/Trigonometric Identities

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