Question

someone please help
if x+1/x=2cosa then prove x^n+(1/x)^n=2cosna
please prove this using de moivre's theorem(complex number)

Answer 1

First one can show that |x| =1
\[
\alpha = 2 \cos(a),\quad -2 \le \alpha \le 2 \\
x +\frac 1 x = \alpha\\
x^2 - \alpha \ x+1 =0\\

x= \frac {\alpha \pm i\ \sqrt{4 -\alpha^2}}{2}\\
|x|^2 = \frac {\alpha^2}{4} + \frac { 4 -\alpha^2}4=1\\
|x| =1
\]

Answer 2

So
\[
x= \cos(t) + i \sin(t)\\
\frac 1 x=\cos(t) - i \sin(t)\\\\
x + \frac 1 x = 2 \cos(a)\\
2 \cos(t) = 2 \cos(a)\\
t= a
\]

Answer 3

\[
x^n = \cos( nt) + i \sin(n t)\\
\frac 1 {x^n}= \cos( nt) - i \sin(n t)\\
x^n+ \frac 1 {x^n}= 2 \cos( n t)= 2 \cos(na)
\]