In 1697, De Moivre was elected into the elite Royal Society. He worked with prominent mathematicians, scientists, and astronomers including Sir Issac Newton and Edmond Halley. De Moivre intense study of mathematics led up to the discovery of new theorem that linked complex numbers to trigonometry. The theory states that for any real number x and any integer n, (cos(x)+ i sin(x))n = cos(nx) + i sin(nx)(about.com). As a result numerous trigonometric problems are able to be obtained with the application of de Moivre's theorem.
Notice he specifically multiplies the i sin(nx) times About.Com!!!! Amazing prediction!