Trying to find the general solution to this homogeneous difference equation:
$$y_k - 2\cos\theta y_{k-1} + y_{k-2} = 0.$$
The characteristic equation is
$$\lambda^2 - 2\cos\theta \lambda + 1 = 0.$$
Not sure how to factor this, but tried
$$(\lambda - \cos\theta)(\lambda - \cos\theta) = 0$$
but I am stuck as to how to get $\cos^2\theta = 1$ using trigonometric identities.
By using the quadratic formula I get a discriminant of
$$4(\cos^2\theta - 1)$$
and I am stuck on how to simplify this to get the general solution.
Any help is appreciated. This is not for homework, but self study.