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

\sec \theta ={\frac {1}{\cos \theta }},\quad \csc \theta ={\frac {1}{\sin \theta }},\quad \cot \theta ={\frac {1}{\tan \theta }}={\frac {\cos \theta }{\sin \theta }}.

{\begin{aligned}\sin \theta &=\pm {\sqrt {1-\cos ^{2}\theta }},\\\cos \theta &=\pm {\sqrt {1-\sin ^{2}\theta }}.\end{aligned}}

{\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta \quad {\text{and}}\quad 1+\cot ^{2}\theta =\csc ^{2}\theta .}

{\displaystyle \sin(2\theta )=2\sin \theta \cos \theta ={\frac {2\tan \theta }{1+\tan ^{2}\theta }}}

\cos(2\theta )=\cos ^{2}\theta -\sin ^{2}\theta =2\cos ^{2}\theta -1=1-2\sin ^{2}\theta ={\frac {1-\tan ^{2}\theta }{1+\tan ^{2}\theta }}

\tan(2\theta )={\frac {2\tan \theta }{1-\tan ^{2}\theta }}

\cot(2\theta )={\frac {\cot ^{2}\theta -1}{2\cot \theta }}

{\displaystyle \sin(3\theta )=-4\sin ^{3}\theta +3\sin \theta }

{\displaystyle \cos(3\theta )=4\cos ^{3}\theta -3\cos \theta }

\tan(3\theta )={\frac {3\tan \theta -\tan ^{3}\theta }{1-3\tan ^{2}\theta }}

\cot(3\theta )={\frac {3\cot \theta -\cot ^{3}\theta }{1-3\cot ^{2}\theta }}

{\displaystyle \sec(3\theta )={\frac {\sec ^{3}\theta }{4-3\sec ^{2}\theta }}}

{\displaystyle \csc(3\theta )={\frac {\csc ^{3}\theta }{3\csc ^{2}\theta -4}}}