Trigonometry/Identities/Table

Reciprocal

 ${\displaystyle \csc \theta ={\frac {1}{\sin \theta }}}$ ${\displaystyle \sec \theta ={\frac {1}{\cos \theta }}}$ ${\displaystyle \cot \theta ={\frac {1}{\tan \theta }}}$

Quotient

 ${\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}}$ ${\displaystyle \cot \theta ={\frac {\cos \theta }{\sin \theta }}}$

Pythagorean

 ${\displaystyle \tan ^{2}\theta +1=\sec ^{2}\theta }$ ${\displaystyle 1+\cot ^{2}\theta =\csc ^{2}\theta }$ ${\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1}$

Negative-angle

 ${\displaystyle \sin(-\theta )=-\sin \theta }$ ${\displaystyle \cos(-\theta )=\cos \theta }$ ${\displaystyle \tan(-\theta )=-\tan \theta }$

Sum and difference

 ${\displaystyle \sin(A+B)=\sin A\cos B+\cos A\sin B}$ ${\displaystyle \sin(A-B)=\sin A\cos B-\cos A\sin B}$ ${\displaystyle \tan(A+B)={\frac {\tan A+\tan B}{1-\tan A\tan B}}}$ ${\displaystyle \cos(A+B)=\cos A\cos B-\sin A\sin B}$ ${\displaystyle \cos(A-B)=\cos A\cos B+\sin A\sin B}$ ${\displaystyle \tan(A-B)={\frac {\tan A-\tan B}{1+\tan A\tan B}}}$

Cofunction

 ${\displaystyle \cos(90^{o}-\theta )=\sin \theta }$ ${\displaystyle \sec(90^{o}-\theta )=\csc \theta }$ ${\displaystyle \cot(90^{o}-\theta )=\tan \theta }$ ${\displaystyle \sin(90^{o}-\theta )=\cos \theta }$ ${\displaystyle \csc(90^{o}-\theta )=\sec \theta }$ ${\displaystyle \tan(90^{o}-\theta )=\cot \theta }$

Double-angle

 ${\displaystyle \cos 2A=\cos ^{2}A-\sin ^{2}A}$ ${\displaystyle \cos 2A=2\cos ^{2}A-1}$ ${\displaystyle \cos 2A=1-2\sin ^{2}A}$ ${\displaystyle \sin 2A=2\sin A\cos A}$ ${\displaystyle \tan 2A={\frac {2\tan A}{1-\tan ^{2}A}}}$

Half-angle

 ${\displaystyle \tan {\frac {A}{2}}=\pm {\sqrt {\frac {1-\cos A}{1+\cos A}}}}$ ${\displaystyle \tan {\frac {A}{2}}={\frac {\sin A}{1+\cos A}}}$ ${\displaystyle \tan {\frac {A}{2}}={\frac {1-\cos A}{\sin A}}}$ ${\displaystyle \cos {\frac {A}{2}}=\pm {\sqrt {\frac {1+\cos A}{2}}}}$ ${\displaystyle \sin {\frac {A}{2}}=\pm {\sqrt {\frac {1-\cos A}{2}}}}$