### 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**.

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