Multiple Angle Formulas for Sine and Cosine
Mathematica can directly produce formulas for the sines and cosines of multiple angles.[more]
Here is a derivation of these formulas. Euler's formula is . Then, , but also, . The relation is known as de Moivre's theorem.
For the case , the right-hand side can be expanded to give
using . Equating the imaginary and real parts leads to the double angle formulas
The cases are similar, using a binomial expansion.[less]