• ${\displaystyle \tan(x)={\frac {\sin(x)}{\cos(x)}}}$
• ${\displaystyle \sec(x)={\frac {1}{\cos(x)}}}$
• ${\displaystyle \cot(x)={\frac {\cos(x)}{\sin(x)}}={\frac {1}{\tan(x)}}}$
• ${\displaystyle \csc(x)={\frac {1}{\sin(x)}}}$

• ${\displaystyle \sin ^{2}(x)+\cos ^{2}(x)=1}$
• ${\displaystyle 1+\tan ^{2}(x)=\sec ^{2}(x)}$
• ${\displaystyle 1+\cot ^{2}(x)=\csc ^{2}(x)}$

• ${\displaystyle \sin(2x)=2\sin(x)\cdot \cos(x)}$
• ${\displaystyle \cos(2x)=\cos ^{2}(x)-\sin ^{2}(x)=1-2\sin ^{2}(x)=2\cos ^{2}(x)-1}$
• ${\displaystyle \tan(2x)={\frac {2\tan(x)}{1-\tan ^{2}(x)}}}$
• ${\displaystyle \cos ^{2}(x)={\frac {1+\cos(2x)}{2}}}$
• ${\displaystyle \sin ^{2}(x)={\frac {1-\cos(2x)}{2}}}$

${\displaystyle \sin(x\pm y)=\sin(x)\cos(y)\pm \sin(y)\cos(x)}$

${\displaystyle \cos(x\pm y)=\cos(x)\cos(y)\mp \sin(x)\sin(y)}$

${\displaystyle \sin(x)\pm \sin(y)=2\sin \left({\frac {x\pm y}{2}}\right)\cos \left({\frac {x\mp y}{2}}\right)}$

${\displaystyle \cos(x)+\cos(y)=2\cos \left({\frac {x+y}{2}}\right)\cos \left({\frac {x-y}{2}}\right)}$

${\displaystyle \cos(x)-\cos(y)=-2\sin \left({\frac {x+y}{2}}\right)\sin \left({\frac {x-y}{2}}\right)}$

${\displaystyle \tan(x)\pm \tan(y)={\frac {\sin(x\pm y)}{\cos(x)\cos(y)}}}$

${\displaystyle \cot(x)\pm \cot(y)=\pm {\frac {\sin(x\pm y)}{\sin(x)\sin(y)}}}$

${\displaystyle \sin(x)\sin(y)={\frac {\cos(x-y)-\cos(x+y)}{2}}}$

${\displaystyle \cos(x)\cos(y)={\frac {\cos(x+y)+\cos(x-y)}{2}}}$

${\displaystyle \sin(x)\cos(y)={\frac {\sin(x+y)+\sin(x-y)}{2}}}$

${\displaystyle \cos(x)\sin(y)={\frac {\sin(x+y)-\sin(x-y)}{2}}}$