Signals and Systems/Table of Trigonometric Identities

Useful Mathematical Identities

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