Trigonometry/Identities/Table

Reciprocal

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

$\cot \theta ={\frac {1}{\tan \theta }}$

Quotient

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

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

Pythagorean

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

$\sin ^{2}\theta +\cos ^{2}\theta =1$

Negative-angle

$\sin(-\theta )=-\sin \theta$ $\cos(-\theta )=\cos \theta$

$\tan(-\theta )=-\tan \theta$

Sum and difference

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

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

Cofunction

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

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

Double-angle

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

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

Half-angle

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

$\cos {\frac {A}{2}}=\pm {\sqrt {\frac {1+\cos A}{2}}}$ $\sin {\frac {A}{2}}=\pm {\sqrt {\frac {1-\cos A}{2}}}$ See also Trigonometry