Math problem #192 (Solution)

Solution to Math problem #192:

Let us employ the method of mathematical induction. The statement is obviously true for $n=1$ and $n=2$. Next, let

$$a_{k-2} = \cos (k-2)\theta, \quad a_{k-1}=\cos (k-1)\theta.$$

Then,

$$a_k = 2\cos\theta\cos (k-1)\theta - \cos (k-2)\theta = \cos k\theta,$$

where we have made use of the trigonometric formula TF21.