The formula of de Moivre Prove for any integer n cos theta

The formula of de Moivre. Prove, for any integer n, (cos theta + i sin theta)^n = cos n theta + i sin n theta. Thus, the angle 3 theta may be treated in terms of theta. Obtain a similar formula for sin 3 theta.

Solution

We can prove the theorem in two way one by Induction method and another by using the expansion of e^_i . Let use the nth power of e^_i

We have eⁱ = Cos + i Sin So eⁿⁱ =(Cos + i Sin )ⁿ

So, eⁿⁱ = 1 + ni + (ni)²/2!   +(ni)³/3! +(ni)⁴/4!+...........

= 1 + i n - n²²/2! - i n³³/3! + n⁴⁴/4! +.............. since i² = -1, i³ = -i, i⁴ = 1 and so on

=(1 - n²²/2! + n⁴⁴/4! - ..........) + i ( n - n³³/3! +......)

= cosn + sinn

So (Cos + i Sin )ⁿ = cosn + sinn

(b) Using n = 3 and expanding we get the result and seperating the real and imoginary, we the part (b) and (c)