Signals and Systems/Table of Trigonometric Identities

Useful Mathematical Identities

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

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