trigonometry - Prove by mathematical induction, or otherwise, that for all integers $nge 1$

$$\cos(1)+\cos(2)+\ldots+\cos(n-1)= \cos(n)-\cos(n-1)/(2\cos(1)-2 ) -1/2$$

Here is my attempt:

Let $P(n)$ be this statement.

$P(1)$ is true since $0=\cos(1)-\cos(1-1)/(2\cos(1)-2) -1/2$

Suppose $P(k)$ is true for some integer $k$. Then I have to prove $P(k+1)$ is also true. That is :

$$\cos(1)+\cos(2)+\ldots+\cos(k-1)+\cos(k)=\cos(k+1)-\cos(k)/(2\cos(1)-2) -1/2.$$

By inductive hypothesis, we have $\cos(k)-\cos(k-1)/(2\cos(1)-2 ) -1/2 +\cos(k)$. But how does it equal to the right hand side? Can someone help me with this question please, thank you!