Need to convert $-mw^2=C(e^{ika}-2+e^{-ika})$ to $w=2\sqrt{\frac{C}{m}}\sin\frac{ka}{2}$

Question

I need to conclude from

$-mw^2=C(e^{ika}-2+e^{-ika})$

that

$w=2\sqrt{\dfrac{C}{m}}\sin\dfrac{ka}{2}.$

Is that possible?

($m, v, k,$ and $ a$ are just some constants -- don't mind them.)

Answer 1

Hint: $$\mathrm e^{ika}+\mathrm e^{-ika}-2=\Bigl(\mathrm e^\tfrac{ika}2-\mathrm e^\tfrac{-ika}2\Bigr)^{\!2}=\Bigl(2i\sin\frac{ka}2\Bigr)^{\!2}.$$