Proving the Relation for Integer n > 1

[Char. Limit]

Homework Statement

I would like to prove the following relation that seems to be true for all integer n > 1.

$$cos(\pi 2^{-n}) = \frac{1}{2} \underbrace{\sqrt{2+\sqrt{2+\sqrt{2+...\sqrt{2}}}}}_{n-1}$$

Homework Equations

I don't really know here.

The Attempt at a Solution

I don't know how to prove this kind of stuff at all, actually. I'm not even certain if this is true, but it certainly seems to be true for small integer n. Can someone help?

 

[losiu99]

Are you familiar with half-angle formula for cosine?

 

[Char. Limit]

I assume you're referring to this?

$$cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1+cos(\theta)}{2}$$

 

[losiu99]

Yes, that's what I was talking about. Can you see how to use it to prove your formula by induction?

 

[Char. Limit]

Ah, yes. I can now. Thanks!