Triangle Identities

In addition to the Law of Sines, the Law of Cosines, and the Law of Tangents, there are numerous other identities that apply to the three angles A, B, and C of any triangle (where A+B+C=180° and each of A, B, and C is greater than zero). Some of the most notable ones follow:

${\displaystyle \cos ^{2}(A)+\cos ^{2}(B)+\cos ^{2}(C)+2\cos(A)\cos(B)\cos(C)=1}$

${\displaystyle \sin(A)+\sin(B)+\sin(C)=4\cos \left({\frac {A}{2}}\right)\cos \left({\frac {B}{2}}\right)\cos \left({\frac {C}{2}}\right)}$

${\displaystyle \tan(A)+\tan(B)+\tan(C)=\tan(A)\tan(B)\tan(C)}$

${\displaystyle \tan \left({\frac {A}{2}}\right)\tan \left({\frac {B}{2}}\right)+\tan \left({\frac {B}{2}}\right)\tan \left({\frac {C}{2}}\right)+\tan \left({\frac {C}{2}}\right)\tan \left({\frac {A}{2}}\right)=1}$

${\displaystyle \cot(A)\cot(B)+\cot(B)\cot(C)+\cot(C)\cot(A)=1}$

${\displaystyle \cot \left({\frac {A}{2}}\right)\cot \left({\frac {B}{2}}\right)\cot \left({\frac {C}{2}}\right)=\cot \left({\frac {A}{2}}\right)+\cot \left({\frac {B}{2}}\right)+\cot \left({\frac {C}{2}}\right)}$

${\displaystyle \sin(A)\sin(B)\sin(C)={\frac {1}{{\bigl (}\cot(A)+\cot(B){\bigr )}{\bigl (}\cot(B)+\cot(C){\bigr )}{\bigl (}\cot(C)+\cot(A){\bigr )}}}}$

${\displaystyle {\frac {\sin(A)+\sin(B)-\sin(C)}{\sin(A)+\sin(B)+\sin(C)}}=\tan \left({\frac {A}{2}}\right)\tan \left({\frac {B}{2}}\right)}$

Pythagoras

${\displaystyle \sin ^{2}(x)+\cos ^{2}(x)=1}$

${\displaystyle 1+\tan ^{2}(x)=\sec ^{2}(x)}$

${\displaystyle 1+\cot ^{2}(x)=\csc ^{2}(x)}$

These are all direct consequences of Pythagoras's theorem.

Sum/Difference of angles

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

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

${\displaystyle \tan(x\pm y)={\frac {\tan(x)\pm \tan(y)}{1\mp \tan(x)\tan(y)}}}$

Product to Sum

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

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

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

Sum and difference to product

${\displaystyle A\sin(x)+B\cos(x)=C\sin(x+y)}$ , where ${\displaystyle C={\sqrt {A^{2}+B^{2}}}}$ and ${\displaystyle y=\pm \arctan {\bigl (}{\tfrac {B}{A}}{\bigr )}}$

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

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

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

Multiple angle

${\displaystyle \cos(2x)=\cos ^{2}(x)-\sin ^{2}(x)=2\cos ^{2}(x)-1=1-2\sin ^{2}(x)}$

${\displaystyle \sin(2x)=2\sin(x)\cos(x)}$

${\displaystyle \tan(2x)={\frac {2\tan(x)}{1-\tan ^{2}(x)}}}$

${\displaystyle \cot(2x)={\frac {\cot(x)-\tan(x)}{2}}}$

${\displaystyle \csc(2x)={\frac {\cot(x)+\tan(x)}{2}}}$

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

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

${\displaystyle \tan(3x)={\frac {3\tan(x)-\tan ^{3}(x)}{1-3\tan ^{2}(x)}}}$

${\displaystyle \cos(4x)=8\cos ^{4}(x)-8\cos ^{2}(x)+1}$

${\displaystyle \sin(4x)=4\sin(x)\cos ^{3}(x)-4\sin ^{3}(x)\cos(x)}$

${\displaystyle \sin ^{2}(4x)=16{\Big [}\sin ^{2}(x)-5\sin ^{4}(x)+8\sin ^{6}(x)-4\sin ^{8}(x){\Big ]}}$

${\displaystyle \tan(4x)={\frac {4\tan(x)-4\tan ^{3}(x)}{1-6\tan ^{2}(x)+\tan ^{4}(x)}}}$

${\displaystyle \cos(5x)=16\cos ^{5}(x)-20\cos ^{3}(x)+5\cos(x)}$

${\displaystyle \sin(5x)=16\sin ^{5}(x)-20\sin ^{3}(x)+5\sin(x)}$

${\displaystyle \tan(5x)={\frac {5\tan(x)-10\tan ^{3}(x)+\tan ^{5}(x)}{1-10\tan ^{2}(x)+5\tan ^{4}(x)}}}$

${\displaystyle \cos(6x)=32\cos ^{6}(x)-48\cos ^{4}(x)+18\cos ^{2}(x)-1}$

${\displaystyle \cos(7x)=64\cos ^{7}(x)-112\cos ^{5}(x)+56\cos ^{3}(x)-7\cos(x)}$

${\displaystyle \sin(7x)=-64\sin ^{7}(x)+112\sin ^{5}(x)-56\sin ^{3}(x)+7\sin(x)}$

${\displaystyle \cos(8x)=128\cos ^{8}(x)-256\cos ^{6}(x)+160\cos ^{4}(x)-32\cos ^{2}(x)+1}$

${\displaystyle \cos(nx)=2\cos(x)\cos {\bigl (}(n-1)x{\bigr )}-\cos {\bigl (}(n-2)x{\bigr )}}$

${\displaystyle \sin(nx)=2\cos(x)\sin {\bigl (}(n-1)x{\bigr )}-\sin {\bigl (}(n-2)x{\bigr )}}$

These are all direct consequences of the sum/difference formulae

Half angle

${\displaystyle \cos \left({\frac {x}{2}}\right)=\pm {\sqrt {\frac {1+\cos(x)}{2}}}}$

${\displaystyle \sin \left({\frac {x}{2}}\right)=\pm {\sqrt {\frac {1-\cos(x)}{2}}}}$

${\displaystyle \tan \left({\frac {x}{2}}\right)={\frac {1-\cos(x)}{\sin(x)}}={\frac {\sin(x)}{1+\cos(x)}}=\pm {\sqrt {\frac {1-\cos(x)}{1+\cos(x)}}}}$

${\displaystyle \cos ^{2}\left({\frac {3x}{2}}\right)=2\cos ^{3}(x)-{\frac {3\cos(x)+1}{2}}}$

In cases with ${\displaystyle \pm }$ , the sign of the result must be determined from the value of ${\displaystyle {\frac {x}{2}}}$ . These derive from the ${\displaystyle \cos(2x)}$ formulae.

Power Reduction

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

${\displaystyle \cos ^{2}(x)={\frac {1+\cos(2x)}{2}}}$

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

Even/Odd

${\displaystyle \sin(-x)=-\sin(x)}$

${\displaystyle \cos(-x)=\cos(x)}$

${\displaystyle \tan(-x)=-\tan(x)}$

${\displaystyle \csc(-x)=-\csc(x)}$

${\displaystyle \sec(-x)=\sec(x)}$

${\displaystyle \cot(-x)=-\cot(x)}$

Calculus

${\displaystyle {\frac {d}{dx}}{\big [}\sin(x){\big ]}=\cos(x)}$

${\displaystyle {\frac {d}{dx}}{\big [}\cos(x){\big ]}=-\sin(x)}$

${\displaystyle {\frac {d}{dx}}{\big [}\tan(x){\big ]}=\sec ^{2}(x)}$

${\displaystyle {\frac {d}{dx}}{\big [}\sec(x){\big ]}=\sec(x)\tan(x)}$

${\displaystyle {\frac {d}{dx}}{\big [}\csc(x){\big ]}=-\csc(x)\cot(x)}$

${\displaystyle {\frac {d}{dx}}{\big [}\cot(x){\big ]}=-\csc ^{2}(x)}$