Assignment:
Moivre-Laplace formula
exp(ix) = cos(x) + i sin(x),
where i = (-1)^(1/2) , and which is widely used in different items of mathematics is usually deduced from the Maclaurin expansions of the functions involved.
But the theory of Taylor (Maclaurin) expansions is a part of more general theory developed in the course of the functions of complex variable. As the Moivre-Laplace formula has numerous applications outside this theory, it seems reasonable to deduce it without references to Maclaurin series.
Q1. To prove the Moivre-Laplace formula: exp(ix) = cos(x) + i sin(x) without use of the Maclaurin expansions.
Provide complete and step by step solution for the question and show calculations and use formulas.