Xem thêm Định lý Ptolemy

Cách chứng minh nhanh các công thức này là dùng công thức Euler.

sin ⁡ ( x ± y ) = sin ⁡ ( x ) cos ⁡ ( y ) ± cos ⁡ ( x ) sin ⁡ ( y ) {\displaystyle \sin(x\pm y)=\sin(x)\cos(y)\pm \cos(x)\sin(y)\,} cos ⁡ ( x ± y ) = cos ⁡ ( x ) cos ⁡ ( y ) ∓ sin ⁡ ( x ) sin ⁡ ( y ) {\displaystyle \cos(x\pm y)=\cos(x)\cos(y)\mp \sin(x)\sin(y)\,} tan ⁡ ( x ± y ) = tan ⁡ ( x ) ± tan ⁡ ( y ) 1 ∓ tan ⁡ ( x ) tan ⁡ ( y ) {\displaystyle \tan(x\pm y)={\frac {\tan(x)\pm \tan(y)}{1\mp \tan(x)\tan(y)}}}   c o t ( x ± y ) = 1 ∓ tan ⁡ ( x ) tan ⁡ ( y ) tan ⁡ ( x ) ± tan ⁡ ( y ) {\displaystyle \ cot(x\pm y)={\frac {1\mp \tan(x)\tan(y)}{\tan(x)\pm \tan(y)}}} c ı s ( x + y ) = c ı s ( x ) c ı s ( y ) {\displaystyle {\rm {c\imath s}}(x+y)={\rm {c\imath s}}(x)\,{\rm {c\imath s}}(y)} c ı s ( x − y ) = c ı s ( x ) c ı s ( y ) {\displaystyle {\rm {c\imath s}}(x-y)={{\rm {c\imath s}}(x) \over {\rm {c\imath s}}(y)}}

với

c ı s ( x ) = e ı x = cos ⁡ ( x ) + ı sin ⁡ ( x ) {\displaystyle {\rm {c\imath s}}(x)=e^{\imath x}=\cos(x)+\imath \sin(x)\,}

và

ı = − 1 . {\displaystyle \imath ={\sqrt {-1}}.\,}