de Moivre's Identity

(1)

From the Euler formula it follows that

(2)

A similar identity holds for the hyperbolic functions,

(3)

REFERENCES:

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 356-357, 1985.

Courant, R. and Robbins, H. What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 96-100, 1996.

Nagell, T. Introduction to Number Theory. New York: Wiley, p. 156, 1951.