trigonometry - Prove that $cosalpha+cos2alpha+cdots+cos nalpha=frac{1}{2}left(frac{sinleft(n+frac{1}{2}right)alpha}{sinfrac{1}{2}alpha}-1right)$

I have to prove using mathematical induction that:
$$\cos\alpha+\cos2\alpha+\cdots+\cos n\alpha=\frac{1}{2}\left(\frac{\sin\left(n+\frac{1}{2}\right)\alpha}{\sin\frac{1}{2}\alpha}-1\right)$$

If I substitute n equals one then I'm giving a such thing as:
$$\cos\alpha=\frac{1}{2}\left(\frac{\sin\frac{3}{2}\alpha}{\sin\frac{1}{2}\alpha}-1\right)$$
But I don't what I should do to prove nextly and that this equation is completed for n+1.