My book's throwing equations out of nowhere

[VietDao29]

My book's throwing equations out of nowhere!

On proving one theorem, my book says:
$$1 + 2\sum_{k = 1} ^ {n} \cos k \alpha = \frac{\sin \left( n + \frac{1}{2} \right) \alpha}{\sin \frac{\alpha}{2}}$$
I have no idea where that formula comes from , and I tried in vain proving it, but I failed. Can anybody gives me a hint?
Thank you,
Any help will be appreciated,
Viet Dao,

 

[Hurkyl]

How did you try? If I remember, it's pretty easy.

 

[Tide]

Think de Moivre! :)

 

[lurflurf]

$$2\sin\left(\frac{a}{2}\right)\cos(k a)=\sin\left(k a+\frac{a}{2}\right) \ - \ \sin\left(k a-\frac{a}{2}\right)$$
so
$$\cos(k a)=\frac{\sin\left(k a+\frac{a}{2}\right) \ - \ \sin\left(k a-\frac{a}{2}\right)}{2\sin\left(\frac{a}{2}\right)}$$

 

[VietDao29]

Thanks, After hours of struggling, I realized that it could be proved by induction. Not very hard, though.
And I also tried lurflurf's way. Thanks.
Thanks a lot,
But... I still don't know how to prove it using de Moivre's formula...
$$(\cos x + i\sin x) ^ n = \cos (nx) + i \sin (nx)$$
How can I do from there?
Viet Dao,

 

[Hurkyl]

Note that cos(nz) = Re[ cis(z)^n ]

( cis(z) := cos z + i sin z = exp(z) )

 

[VietDao29]

Thanks a lot, guys.
Viet Dao,