# Complex number proof involving trigonometry

Let$$z$$ be a complex number such that $$z + \frac{1}{z} = \cos(x)$$ Then what is the value of the expression $$z^n + \frac{1}{z^n}$$ where $$n$$ is an integer?

Please help me, i have tried somehow using the trigonometric way of defining complex numbers but still didn't manage to get anywhere.

• What do you mean by $z+\frac{1}{z}=\cos(x)?$ – Chickenmancer Mar 14 at 18:58
• Is $z=x+iy$ and the $x$ in $\cos(x)$ refers to that?? – Anurag A Mar 14 at 19:01
• z+1/z = cos(x) it's a relation that needs to help us to get to the other answer. – Melinceanu Victor Mar 14 at 19:04
• so what is $x$? How is it related to $z$? – Anurag A Mar 14 at 19:07
• You know, this would be a whole lot nicer if that were $z+\frac1z=2\cos x$... – jmerry Mar 14 at 19:46

Let $$z = \cos x + i \sin x$$ and $$\dfrac {1}{z} = \cos x - i \sin x$$
Then by DeMoivre's theorem $$z^n = (\cos x + i \sin x)^n = \cos nx + i \sin nx$$ $$\dfrac {1}{z^n} = (\cos x - i \sin x)^n = \cos nx - i \sin nx$$
What do you need to get $$z^n + \dfrac {1}{z^n}?$$
• Here $z+\frac 1z=2\cos x$ but OP said $z+\frac 1 z=\cos x$ – J. W. Tanner Mar 14 at 23:01