Ta có: \(\cos 4\varphi + i\sin 4\varphi = {\left( {\cos \varphi + i\sin \varphi } \right)^4}\)
\(\eqalign{ & = {\cos ^4}\varphi + 4\left( {{{\cos }^3}\varphi } \right)\left( {i\sin \varphi } \right) + 6\left( {{{\cos }^2}\varphi } \right)\left( {{i^2}} \right){\sin ^2}\varphi + 4\left( {\cos \varphi } \right)\left( {{i^3}{{\sin }^3}\varphi } \right) + {i^4}{\sin ^4}\varphi \cr & = {\cos ^4}\varphi - 6{\cos ^2}\varphi {\sin ^2}\varphi + {\sin ^4}\varphi + \left( {4{{\cos }^3}\varphi \sin \varphi - 4\cos \varphi {{\sin }^3}\varphi } \right)i. \cr} \)
Từ đó: \(\cos 4\varphi = {\cos ^4}\varphi - 6{\cos ^2}\varphi {\sin ^2}\varphi + {\sin ^4}\varphi \)
\(\sin 4\varphi = 4{\cos ^3}\varphi \sin \varphi - 4\cos \varphi {\sin ^3}\varphi \)
