Prove $\cos\frac{\pi}{2^{n+1}}$ is irrational

Question

Prove that for every number $n\in\mathbb N$,number $\cos\frac{\pi}{2^{n+1}}$ is irrational. I really don't know where to start.

Answer 1

Suppose $\cos(\pi/2^{n+1})$ is irrational, but $\cos(\pi/2^{n+2})$ is rational; set $\alpha=\pi/2^{n+2}$; then $2\alpha=\pi/2^{n+1}$ and $$ \cos2\alpha=2\cos^2\alpha-1 $$

Can you see the contradiction?

To finish, note that for $n=1$ we have $\cos(\pi/4)=\sqrt{2}/2$.

Answer 2

Proof by mathematic induction:

base case:

when $n = 1, \cos({\frac{\pi}{2^{n + 1}}}) = 2\sqrt{2}$ is clearly irrational

Now suppose when $n = k, \cos({\frac{\pi}{2^{k + 1}}})$ is irrational

When $n = k + 1$

\begin{align} \cos({\frac{\pi}{2^{k + 2}}}) &=\cos({\frac{\pi}{2^{k + 1}}} \times\frac{1}{2})\\ &=\sqrt{\frac{\cos({\frac{\pi}{2^{k + 1}}}) + 1}{2}} \end{align}

$\because$ an irrational number plus a rational number is irrational, the constant multiple of irrational number is irrational, and power of irrational number is irrational

$\therefore$ when $n = k + 1$ the original proposal is true

$\therefore$ the statement is proved.